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•  • 


WARREN'S  COMPLETE  COURSE 


IN 


Descriptive  Geometry,  Stereotomy  and  Drawing, 

FOR  SCHOOLS.  COLLEGES,  POLYTECHNIC  INSTITUTES. 

Published  by  JOHN  WILEY  &  SONS, 

53    East   Tenth    Street,    New  York. 


1.  A  PRIMARY  GEOMETRY  for  CoifMox  Schools. 

WITH  SIMPLE  AXD  PRACTICAL  EXAMPLES  /Y  PLAJSfE  AXD  PROJEC- 
TION DRAWLS  Q,  AND  SUITED  TO  ALL  BEGINNERS. 

By  S.  Edward  Warren,  C.E., 

Author  of  a  Series  of  Elementary  and  Higher  Text-books  on  the  Principles  and 
Practice  of  Industrial  Ih-awing. 

12mo,  Cloth.    75  Cents. 

There  is  a  growing  and  well-grounded  feeling  that  there  is  an 
excess  of  attention  to  hook  and  brain  education,  as  compared  with 
hand  and  brain  education. 

In  the  present  increasing  reaction  from  the  old  abandoned 
apprenticeship  system,  this  feeling  manifests  itself  in  a  demand  for 
increasing  attention  to  Manual  Training. 

Whether  this  manual  training  shall  be  in  schools,  or  in  real  life, 
in  shops,  ships,  railroads,  ofl&ces  or  fields,  Geometry,  or  the  study 
oiform,  should,  from  the  beginning,  accompany  arithmetic,  or  the 
study  of  numler;  since  all  mechanical  industries,  of  every  name 
and  kind  and  grade,  require  familiarity  with  forms  and  their 
ajrangement. 

Nearly  all  the  Geometries  hitherto  published,  and  styled  ''Ele- 
mentary," differ  from  the  larger  ones,  suited  to  colleges,  more  in 
size  than  in  method,  and  are  seldom  begun  before  entering  the 
High  School;  thus  depriving  many  thousands  who  need  it,  of  an 
opportunity  to  study  geometry. 


WARREN'S  FBIMART  GEOMETRY, 

Prepared  as  the  result  of  much  experience  and  reflection,  is,  as 
indicated  by  its  title,  much  more  elementary  than  those  commonly 
80  called ;  as  may  be  gathered  from  the  following  statement  of 

Its  Distinctive  Features. 

1.  It  presents  the  simplest  geometrical  truths,  at  first  in  a  famil- 
iar conversational  style,  and  is  suited  to  the  common  schools  in 
which  so  many,  perhaps  most,  children  end  their  school  days. 

2.  Its  numerous  and  appropriate  illustrations  are  original. 

3.  It  contains  numerous  suitable  examples  for  practice. 

4.  It  embraces  such  a  selection  of  truths  of  for7n,  as  well  as 
measure,  as  to  be  especially  adapted  to  accompany  industrial  draw- 
ing and  other  manual  training. 

5.  It  is  not  encumbered  and  confused  by  embracing  too  little 
of  each  of  too  many  and  diverse  applications. 

SPECIMENS  OF  MANY  COMMENDATIONS. 

"  Geometry  might  be  taught  much  sooner  than  it  is,  and  your  book  is  well  fitted 
to  do  it." — R.  D.  Dodge,  Prin.  Prospect  Birk  Imt.,  Brooklyn,  N.  Y. 

"lam  pleased  with  it  as  a  practical  work,  and  one  that  will  make  it  natural  and 
easy  to  introduce  youth  to  the  study." — D.  L.  Kiehle,  State  SupH  Pub.  Inst.,  Minn. 

"  It  impresses  us  very  favorably.  I  fully  agree  with  what  you  say  in  your  pref- 
nce. — Richard  Edwards,  LL.D.,  State  SupH  Puh.  Inst.,  lU. 

"Prepared  with  great  care.  .  .  .  More  elementary  than  most  Geometries.  .  .  . 
Might  be  used  in  the  grammar  grade  with  good  effect.  .  .  ." — New  England  Journal  of 
Education. 

"  1  like  your  Primary  (Jeometry  very  much." — From  a  recent  letter  from  a  lauj 

Teacher.  

PBOFESSOR  WARREN'S  OTHER  ELEMENTARY  WORKS, 

Suited  to  Academies  and  to  Preparatory,  High,  Normal,  Evening, 
AND  Industrial  or  Manual  Schools  are  as  follows  : 

2.  FREE-HAND  GEOMETRICAL  DRAWING,  widely  and 
variously  useful  in  training  the  eye  and  hand  in  accurate  sketching 
of  plane  and  solid  figures,  lettering,  geometric  beauty,  and  design, 
«tc.     12  folding  plates,  many  cuts.     Large  12mo,  cloth,  $1.00. 


3.  DRAFTING  INSTRUMENTS  AND  OPERATIONS.  A  full 
description  of  drawing  instruments  and  materials,  with  applications 
to  useful  examples;  tile  work,  wall  and  arch  faces,  ovals,  etc.  7 
folding  plates,  many  cuts.     Large  12mo,  cloth,  $1.25. 

4.  ELEMENTARY  PROJECTION  DRAWING.  Fully  explain- 
ing, in  six  divisions,  the  principles  and  practice  of  elementary 
plan  and  elevation  drawing  of  simple  solids;  constructive  details; 
shadows;  isometrical  drawing;  elements  of  machines;  simple  struct- 
ures.   24  folding  plates,  numerous  cuts.     Large  12mo,  cloth,  $1.50. 

"We  know  of  no  manuar  on  the  subject  so  clear  and  full  in  its  statements 
of  principles  and  in  its  solutions  of  problems,  and  so  practical  in  the  objects 
selected  for  study.  It  will  be  of  great  value  to  all  studious  young  artisans,  as 
well  as  for  use  in  academies  and  liigli-scliools. " — New  England  Journal  of 
Education. 

5.  ELEMENTARY  PERSPECTIVE.  With  numerous  practical 
examples,  and  every  step  fully  explained.  Commended  to  Deco- 
rators, Ladies'  Seminaries,  Architects,  etc.  Numerous  cuts.  Re- 
vised edition.     2  plates.     Large  12mo,  cloth,  $1.00. 

6.  PLANE  PROBLEMS  on  the  Point,  Straight  Line,  and  Circle. 
225  problems.  Many  on  Tangencies,  and  other  useful  or  curious 
ones.  Excellent  to  accompany  any  Geometry,  and  with  instru- 
mental constructions.  For  Colleges  and  Polytechnic  Schools ;  and 
for  Architects,  Engineers,  Machinists,  and  Draftsmen.  150  wood- 
cuts, and  plates.     Large  12mo,  cloth,  $1.25. 


HIGHER  WORKS. 

1.  THE  ELEMENTS  OF  DESCRIPTIVE  GEOMETRY,  SHAD- 
OWS AND  PERSPECTIVE,  with  brief  treatment  of  Trihedrals; 
Transversals;  and  Spherical,  Axonometric,  and  Oblique  Projec- 
tions; and  many  examples  for  practice.  The  lest  complete  short 
course;  embracing  everything  which  is,  and  omitting  everything 
which  is  not,  generally  wanted.  24  folding  plates.  8vo,  cloth,  $3.50. 

2.  PROBLEMS,  THEOREMS,  AND  EXAMPLES  IN  DESCRIP- 
TIVE GEOMETRY.  Entirely  distinct  from  the  above,  with  115 
problems,  embracing  many  useful  constructions;  52  theorems,  in- 


eluding  examples  of  the  demonstration  of  geometrical  properties  hy 
the  method  of  projections;  and  many  examples  for  practice.  24 
folding  plates.  Should  form  at  least  an  elective  study  in  the  ge?i- 
eral  mathematical  course  of  every  College;  and  also  well  adapted, 
in  conjunction  with  the  following,  to  Polytechnic  Schools.  8vo, 
cloth,  $2.50. 

3.  GENERAL   PEOBLEMS   IN   SHADES   AND    SHADOWS. 

A  beautiful  and  useful  application  of  Descriptive  Geometry,  consist- 
ing of  the  geometrical  construction  of  Shades  and  Shadows  ;  with 
practical  examples,  including  every  kind  of  surface.  15  folding 
plates.    8vo,  cloth,  $3.00. 

4.  GENERAL  PROBLEMS  IN  THE  LINEAR  PERSPECTIVE 
OF  FORM,  SHADOW,  AND  REFLECTION.  A  complete  treatise 
on  the  principles  and  practice  of  perspective  by  various  older  and 
recent  methods;  in  98  problems,  including  many  practical  ones  of 
an  architectural  character,  24  theorems,  and  with  17  large  plates. 
Detailed  contents,  and  numbered  and  titled  topics  in  the  larger 
problems,  facilitate  study  and  class  use.  Eevised  edition.  Correc- 
tions, changes  and  additions.     8vo,  cloth,  $3. 50. 

5.  ELEMENTS  OF  MACHINE  CONSTRUCTION  AND  DRAW- 
ING. 73  classified  standard  and  broadly  representative  practical 
examples  drawn  to  scale  and  of  great  variety;  besides  30  problems 
and  31  theorems  relating  to  gearing,  belting,  valve-motions,  screw- 
propellers,  etc.  All  fully  ex2)lained  in  the  text.  2  vols.,  8vo,  cloth, 
one  of  text,  one  of  34  folding  plates.     $7.50. 

6.  PROBLEMS  IN  STONE-CUTTING.  20  problems,  with  ex- 
amples for  practice  under  them,  arranged  according  to  the  dominant 
surface  (plane,  developable,  warped  or  double-curved)  in  each,  and 
embracing  every  variety  of  structure;  gateways,  stairs,  arches, 
domes,  winding  passages,  etc.  Elegantly  printed  at  the  Kiverside 
Press.     10  folding  plates.     8vo,  cloth,  $2.50. 

"  Professor  Warren  here  presents  a  most  thorough  and  progressive  course 
in  Drawing,  comprising  Free-hand  Drawing,  Plane  Problems,  Projection 
Drawing,  Linear  Perspective,  and  the  higher  courses  in  the  same,  including, 
among  others.  Machine.  Drawing,  Stone  Cutting,  and  a  new  Descriptive 
Geometry.  Shades  and  Perspective,  a  condensed  treatise,  entirely  new  and 
thoroughly  practical."— iVew  England  Journal  of  Education. 

^*»  Full  descriptive  catalogues  and  circulars,  with  references  and  testi- 
monials, on  application. 


THE 


ELEMENTS 


OP 


Coordinate  Geometry, 

IN 

THREE  PARTS. 

I.   Cartesian  Geometry,       II.  Quaternions, 
III.  Modern  Geometry, 

AND  AN 

APPENDIX 


DE    YOLSON    WOOD, 

Pro»E3S"b  Mathesiatics  and  Mechanics  in  Stevens  iNSTrrtrrE  of 
Technology. 


SEVENTH    EDITION. 


THIRD    THOUSAND. 


NEW  YORK: 

JOHN    WILEY    AND    SONS, 

53  East  Tenth  Street, 

1895. 


Copyright 

BT  Db  VOIiSON  WOODi 


engineering  & 
(j^  p\      Mathematical 
<-  cr  I  Screnees 

^  y  I  Library 


W^i". 


J2^ 


I'l^ 


PREFACE. 


This  book  contains  some  matter  not  heretofore  foiTnd  in 
works  upon  Analytical  Geometry.  As  it  is  designed  as  a 
text-book,  care  lias  been  taken  to  separate  the  different  sub- 
jects so  that  they  may  be  studied  advantageously,  each  by 
itself.  The  Cartesian  system  will  naturally,  if  not  necessa- 
rily, be  studied  first,  for  it  is  not  only  the  most  common,  but 
is  the  leading  system  used  in  the  Calculus.  The  matter  per- 
taining to  the  conic  sections  is  considerably  condensed,  com- 
pared with  most  other  works  which  treat  of  the  subject. 
This  has  been  accomplished  by  treating  of  the  several  curves 
under  one  head  when  discussing  a  property  which  is  com- 
mon to  all  of  them.  By  this  arrangement  we  trust  that  some 
time  will  be  saved  to  the  student  in  this  part  of  the  work, 
and  thus  enable  him  to  give  more  time  to  advanced  portions 
of  the  subject. 

The  subject  of  Quaternions  is  treated  in  the  most  elemen- 
tary manner,  and  the  examples  are  of  the  simplest  kind,  the 
object  being  to  explain  and  illustrate  the  principles  and  the 
character  of  the  operations  without  taxing  the  ingenuity 
of  the  student  in  the  mere  solution  of  problems.  One  cannot 
form  a  correct  judgment  of  the  power  of  this  analysis  from 
these  examples,  but  to  attempt  to  explain  its  higher  processes 
would  be  equivalent  to  excluding  it  from  our  courses  of 
study.  If  the  presentation  here  made  of  the  subject  suc- 
ceeds in  creating  an  interest  in  it,  and  of  establishing  a 
correct  foundation  for  its  future  study,  all  will  be  accom- 
plished that  was  intended.  The  English  works  upon  the 
subject  are  not  numerous.     The  only  ones  known  to  the 

ill 

790074 


iv  PREFACE. 

author  are,  —  Hamilton's  Lectures  upon  Quaternions,  Ham- 
ilton's Elements  of  Quaternions,  Tait's  Quaternions,  and  Kel- 
land  &  Tait's  Introduction  to  Quaternions,  the  last  of  which  is 
probably  the  best  elementary  work  upon  the  subject  hitherto 
published.  There  is,  however,  considerable  literature  upon 
the  subject  in  English  and  other  foreign  journals,  some  of 
which  is  contained  under  the  head  of  discussions  upon  more 
general  systems  of  algebra.  It  is  with  pleasure  that  I  ac- 
knowledge my  indebtedness  to  my  friend  and  former  pupil, 
Mr.  Henry  A.  Beckmeyer,  for  valuable  assistance  in  the  pre- 
paration of  this  part  of  the  work. 

The  third  part  of  the  work,  on  Modern  Geometry,  is  very 
brief,  and  is  intended  chiefly  to  expand  still  further  the  ideas 
of  the  student,  by  showing  that  a  great  variety  of  systems 
may  be  used.  Still,  scarcely  enough  is  given  to  enable  him 
to  raise  the  veil  and  witness  the  scenes  beyond.  For  this  he 
must  consult  other  works.  The  amount  of  literature  upon 
this  subject,  mostly  stored  in  foreign  mathematical  jour- 
nals, is  immense ;  and  those  only  who  give  their  time  to  its 
study  can,  in  any  sense,  become  masters  of  it.  Of  the  Eng- 
lish works  upon  the  subject,  we  notice  Trilinear  Coordinates 
and  3Iethods  of  Modern  Analytical  Geometry,  by  W.  A.  Whit- 
worth,  Salmon's  Conic  Sections,  Trilinear  Coordinates  by  W.  J. 
Wright  (being  No.  2  of  his  Mathematical  Tracts),  Trilinear 
CdfJrdinates,  by  N.  M.  Ferrers.  There  are  other  English,  and 
also  several  French  and  German  works  which  we  have  not 
been  able  to  consult,  which  may  be  as  meritorious  and  possi- 
bly more  extensive  than  those  above  mentioned. 

Several  of  the  cuts  of  Higher  Loci  were  taken — by  per- 
mission of  the  authors — from  Rice  and  Johnson's  Elements 
of  the  Differential  Calculus.  De  V.  W. 

HoBOKBN,  March,  1879- 


CONTENTS. 


[The  numbers  of  the  Articles  are  placed  at  the  head  of  the  page.] 


Pabt  I.— caktesian  geometry. 


CHAPTER  I. 

DEFINITIOIfS— EQUATIONS  TO  A  POINT. 

Definitions :  —  Lines,  when  known  (1)  ;  System  of  coordinates  (2) ; 
Known  point  (3) ;  Right  line  (4)  ;  Illustrations  (5)  ;  Coordinate  axes 
(6) ;  Kinds  of  coSrdinates  (7)  ;  The  ordinate  (8)  ;  Abscissa  (9)  ;  Co- 
ordinates to  a  point  (10)  ;  Equations  to  a  point  (11)  ;  Examples. 
Polar  System  : — Initial  line  (12) ;  The  pole  (13) ;  Radius  vector  (14) ; 
Variable  angle  (15)  ;  Signs  of  the  coordinates  (16>  ;  Polar  coordinates 
of  a  point  (17) ;  Polar  equations  of  a  point  (18) ;  Examples.  Dis- 
tance between  two  Points  : — Bilinear  coordinates  (19) ;  Rectangular 
coordinates  (20) ;  Polar  coordinates  (21) ;  Examples Pages  l-2t 

CHAPTER  II. 

THE  BIGHT  LINE. 

Definitions : — Locus  (22)  ;  Consecutive  points  (23)  ;  Equation  of  a  locus 
(24) ;  Analytical  Geometry  (25)  ;  Coordinate  Geometry  (26).  Eqtui- 
tions  to  a  Right  Line  : — Bilinear  equation  (27)  ;  Rectangular  equa- 
tion (28)  ;  Intercepts  (29)  ;  In  terms  of  perpendicular  (30)  ;  Con- 
stants and  variables  (31)  ;  Absolute  term  (32)  ;  Discussion  (33-35)  ; 
Polar  equation  (36) ;  Examples.  Point  and  Line  (38-40) ;  Ex- 
amples. Of  two  Lines  .-—Intersection  of  (41)  ;  Parallel  to  (42) ; 
Line  through  a  point  (43)  ;  Angle  between  (44)  ;  Perpendicular  to 
g^iven  line  (45, 46) ;  Line  cutting  another  at  a  given  angle  (47) ;  Ex- 
amples. 

CHAPTER   ni. 

TKANSFOKMATION    OP    COORDINATES. 

Formalas  for  passing  from  one  bUinear  system  to  another  (49-51) ; 
Rectangular  to  oblique  coordinates  (52)  ;  Oblique  to  rectangular 
(53) ;  Rectangular  to  rectangular  (64)  ;   Rectangular  to  polar  (55)  ; 

Examples Pages  27-33 

V 


^  CONTENTS. 

CHAPTER  IV. 

CONIC  SECTIONS. 

The  Circle  .-—Definition  (57-59) ;  Discussion  (60) ;  Constniction  of  the 
locus  (61) ;  Discussion  (62) ;  Definition  of  centre  (63)  ;  Diameter 
and  axis  (64) ;  Examples.  The  Ellipse  .-—Definition  (65) ;  To  trace 
the  curve  (66)  ;  Definitions  (67) ;  Equations  of  the  cui-ve  (68-72)  ; 
Examples.  The  Hyperbola  .-—Definition  (73)  ;  Construction  (74, 
75)  ;  Definitions  (76)  ;  A  branch  (77)  ;  Equations  of  the  curve 
(78-81);  Examples.  r/icParofto^a  . •— Definition  (82);  To  construct 
the  curve  (83),  (85)  ;  Definitions  (84) ;  Equations  of  (86-88)  ;  Ex- 
amples. Equations  to  the  Conic  compared  {90,  9\);  General  equa- 
tion of  the  second  degree  (92) ;  Examples  ;  Eccentricity  (93)  ; 
Eccentric  angle  (93a) ;  Latus  rectum  (94,  95)  ;  Remark  (96) ;  Ex- 
amples. Of  Ordinates  (97)  :  Intersections  of  a  right  line  with 
conies  and  conies  with  each  other  (98) ;  Examples.  Of  Tangents  : 
—To  the  ellipse  (100)  ;  Circle  (101)  ;  Hyperbola  (102)  ;  Parabola 
(103);  Intercepts  of  (104);  Examples;  Eccentric  angle  (104^); 
Subtangents  (105) ;  Length  of  tangent  (106) ;  Examples  ;  Angles 
between  focal  radii  and  the  tangent  (107)  ;  Construction  of  a  tan- 
gent to  a  conic  (108) ;  Normal  and  Subnormal  to  Conies  (109-113)  ; 
In  terms  of  eccentric  angle  (ll3a) ;  Intercepts  of  the  normal  (114) ; 
Length  of  subnormal  (114a)  ;  Length  of  normal  (115)  ;  Distance 
of  foot  of  normal  from  either  focus  (116) ;  The  normal  bisects 
the  angle  between  the  focal  radii  (117)  ;  Construction  of  the  nor- 
mal (118);  Examples;  Linear  equation  of  the  conic  (119)  ;  Bos- 
eovich's  Definition  (120)  ;  To  construct  the  hyperbola  (121)  ;  Sup- 
plementary chords  (122,  123)  ;  Conjugate  diameters  (124,  125)  ;  To 
construct  a  tangent  to  a  conic  by  means  of  conjugate  diameters 
(126)  ;  Oblique  axes  (127)  ;  Conjugate  diameters  (128,  129)  ;  Equa- 
tion of  the  tangent  referred  to  conjugate  diameters  (130)  ;  Proper- 
ties of  conjugate  diameters  (131) ;  Parabola  referred  to  oblique  axes 

(132,  133) ;  Interpretation  of  i  ■^■^..^  (134, 135);  Double  focal  ordi- 
nate of  the  parabola  (136) ;  Construction  of  the  Parabola  (131-142) ; 
To  construct  a  tangent  to  a  parabola  parallel  to  a  line  (143)  ;  To 
find  the  axis  of  a  parabola  (144) ;  Parameters  (145,  146) ;  Pole  and 
Polar  :  —  Definitions  (147-149) ;  Equations  of  (150-153)  ;  Polars 
of  special  points  (154,  155)  ;  Examples.  The  Hyperbola  and  its 
Asymptotes  .-- Definition  (156) ;  Equations  of  the  asymptotes  (157) ; 
Equations  of  conditions  for  asymptotes  (158)  ;  Equation  of  hyper- 
bola referred  to  its  asymptotes  (159) ;  <p  =  2  sec-i  e  (160)  ;  Prob- 
lem (161)  ;  Equation  to  any  chord  (162)  ;  Equation  to  the  tangent 
(163)  ;  Intercepts  of  tangent  (164) ;  Intercepts  equal  (165)  ;  Tan- 
gents at  the  extremity  of  conjugate  diameters  meet  on  the  asymp- 
totes (166)  ;  Polar  Equations  of  Conies  : — General  equations  (167)  : 
Of  parabola  (168);  Of  ellipse  (169),  (174);  Of  hyperbola  (170), 
(175) ;  Discussion  (171-173) ;  Examples Pages  34-118 


^ 


/ 


/ 

CONTENTS.  '  vii 

CHAPTER  V.  '         ^ 

GENERAL  DISCUSSION  OF  THE  EQUATION  OF  THE  SECOND  DEGREE. 

Rectangular  equation  to  a  conic  having  any  position  in  the  plane  (176)  ; 
Every  equation  of  the  second  degree  may  represent  a  conic  (177) ; 
General  test  (178)  ;  To  make  xy  disappear  (179)  ;  To  cause  the  co- 
efficient of  y  to  disappear  (180)  ;  Remark  (181)  ;  Varieties  of  the 
ellipse  (182)  ;  Hyperbola  (183);  Parabola  (184);  Illustrations 
(185);  To  pass  a  conic  section  through  five  points  (186);  Exam- 
ples. Another  method  of  discussing  the  equation  of  the  second 
degree  (187)  ;  To  remove  the  term  containing  xy  from  the  general 
equation  (187«)  ;  To  remove  the  terms  containing  the  first  powers 
X  and  y  from  the  equation  (1876)  ;  Examples Pages  119-147 

CHAPTER  VI. 

LOCI    IN    SPACE. 

Of  the  Point,  Right  Line,  and  Plane. 

Definitions  (189)  ;  Axis  of  a  plane  (190)  ;  Order  of  the  angles  (191)  ; 
Coordinates  of  a  point  (192);  Equations  of  the  axes  (1926);  Distance 
between  two  points  (193)  ;  Projections  (194).  The  Right  Line  : — 
Equations  of  (195)  ;  Intersections  of  (196)  ;  Passing  through  a 
point  (197)  ;  Inclination  of  to  the  axes  (198) ;  Angle  between  two 
lines  (199).  The  Plane  .-—Equation  of  (201) ;  Equation  of  first 
degree  (202) ;  Discussion  (203)  ;  In  terms  of  perpendicular  (204, 
205)  ;  Intersection  of  (206)  ;  Passing  through  two  points  (207)  ; 
Inclination  to  the  axes  (208)  ;  Angle  between  planes  (209)  ; 
Parallel  planes  (210) ;  Perpendicular  planes  (211).  Line  and 
Plane  ;— Line  pierces  a  plane  (212)  ;  Line  perpendicular  to  a  plane 
(213-216)  ;  Angle  between  line  and  plane  (217).  Transformation 
of  Coordinates : — To  transform  from  rectangular  to  oblique  axes 
(219) ;  Planar  to  polar  (220) ;  Polar  to  planar  (221) ;  Examples. 

Pages  148-170 
CHAPTER  VII. 

CURVED  SURFACES. 

Definitions  (222).  Of  Cylindrical  Surfaces  (223-225).  Conical  Surfaces 
(226-228).  Sphere  (229)  ;  Surfaces  of  revolution  (230).  Ellip- 
soids (231-233).  Paraboloids  (234-237).  Eypertoloids  (238-241). 
Of  Intersections  (242-252).  Discussion  of  Quadrics  (253-260)  ; 
Tangent  planes  (261,   262)  ;  Of  normals  (263);  Examples. 

Pages  171-201 

CHAPTER  Vin. 

LOCI  OP  HIGHER  ORDERS. 

Definitions  (264).  Spirals  .-—Spiral  of  Archimedes  (266)  ;  Hyperbolic 
spiral  (267) ;    Logarithmic  (268) ;  Involute  (269) ;  Lituiis  (270)  ; 


viii  CONTENTS. 

Parabolic  spiral  (271).  Trigonometrical  Curves  (272-277)  ;  Loga- 
ritlimic  curve  (278)  ;  Parabolas  (279-283)  ;  Trochoids  (284-294) ; 
Conchoid  of  Nicomedes  (295,  296)  ;  Trisection  of  an  angle  (297) ; 
Remark  (298);  The  cissoid  (299-301);  Two  mean  proportionals 
(302)  ;  Duplication  of  a  cube  (303)  ;  Quadratrix  of  Dionstratus  (304, 
306)  ;  Witch  of  Agnisi  (306)  ;  Ovals  (307-311)  ;  Curves  of  pursuit 
(312);  Folium  (313);  Discontinuous  cuives  (314-317);  Loxo- 
dromic  (318) ;  Logocyclic  (319) ;   Helix  (320) ;  Conoid  (321,  322). 

Pages  203-228 


Paet  n.— quateenions. 


CHAPTER  I. 

ADDITIGN  AND  SUBTRACTIOX  OF  VECTORS. 

Definitions  (323) ;  Vectors  (324-328)  ;  Tensor  (329).  Vector  Equa- 
tions : — Equations  and  law  of  signs  (330-333) ;  Examples.  Sum 
of  independent  vectors  separately  equal  zero  (334)  ;  APPLICA- 
TIONS ;  Examples.  Medial  vectors  (336,  337)  ;  APPLICATIONS  ; 
Examples.  Co-planar  vectors  (338-340);  APPLICATIONS;  Ex- 
amples. Angle-bisectors  (341,  342)  ;  APPLICATIONS  ;  Exam- 
ples. Transversals  (343) ;  APPLICATIONS,  (344-349) ;  Com- 
plete quadrilateral  (350) Pages  239-263 

CHAPTER  II. 

MUIiTIPLICATION   AND  DIVISION  OP  VECTORS. 

Notation  (351) ;  Operation  (352-354)  ;  Symbol  V-i  (355)  ;  Non-com 
mutative  principle  (356) ;  Associative  principle  (357) ;  Examples. 
Square  of  a  vector  (359) ;  Division  of  rectangular  vectors  (360) ; 
Examples.  Oblique  Vectors  : — Multiplication  of  (361) ;  Division 
of  (362) ;  Examples  ;  Comparing  results  (363) ;  Scalars  (364) ; 
Versor  (365)  ;    A     QUATERNION   (366)  ;    General  expressions 

20 

(367) ;  Discussion  (368)  ;  Examples,  e"  (369)  ;  Proposition 
(370) ;  Example  ;  Plane  triangles  (371-378)  ;  Three  co-initial  co- 
planar  vectors  (379) ;  Triedrals  (380)  ;  fj  =  ix  +  jy  +  kz  (381) ;  Con- 
jugate quaternions  (382) Pages  264-289 

CHAPTER  III. 

LINE,   PLANE,    SPHERE,   AND  C0NIC8. 

Right  line  (383)  ;  Examples.  Plane  (384) ;  Circle  (385) ;  Equation 
of  tangent  line  (386) ;  Examples.  Sphere  (387)  ;  Ellipse  (388)  ; 
Hyperbola  (389) ;  Parabola  (390) Pages  390-300 


CONTENTS.  ix 

Part  HI.— MODERN  GEOMETRY. 


CHAPTER    I. 

Modem  Geometry,  definition  (391)  ;  Definition  of  locus  (392).  Tan- 
gential System  : — Definition  (393) ;  Equation  of  four  cusp  hypocy- 
cloid  (394);  Equation  of  the  ellipse  (395)  ;  Equation  of  a  point  (396); 
Equation  of  a  line  (397) ;  Examples  ;  Equation  of  a  line  in  terms 
of  three  perpendiculars  from  three  fixed  points  (398).  THlinear 
System  : — Notation  (399) ;  Signs  of  the  perpendiculars  (400)  ;  Re- 
lation between  the  coordinates  (401)  ;  Bisectors  (402)  ;  Equation  of 
a  line  (403) ;  Abridged  notation  (404).  Other  Systems : — List  of 
twenty -two  systems  by  Rev.  Thomas  Hill  (405) ;  Other  systems 
(406,407) Pages  303-316 

APPENDIX  I. 
Brief  sketch  of  the  history  of  Quaternions Pages  317-329 

Ai^PENDIX  II. 
Hyper-Space 330 


GEEEK    ALPHABET. 


Letters. 

Names. 

Letters. 

Names. 

A    a 

Alpha 

N  V 

Nu 

B   /3 

Beta 

H   B, 

Xi 

r  Y 

Gamma 

.0    o 

Omicron 

A    S 

Delta 

77  n 

Pi 

E  e 

Epsilon 

P    p 

Eho 

z  a 

Zeta 

2  G  s: 

Sigma 

H  V 

Eta 

T    T 

Tau 

e  ^  e 

Theta 

T   V 

Upsilon 

I  t 

Iota 

0   (p 

Phi 

K     H 

Kappa 

X  X 

Chi 

A   \ 

Lambda 

W  ip 

Psi 

M  IX 

Mu 

£1  00 

Omega 

PAKT    I. 


CAETESIAJSI'    GEOMETRY. 


COORDINATE    GE03IETRY. 


CHAPTEE  I. 

FUNDAMENTAL  PRINCIPLES. 

Definitions. 

1 .  Lines  are  said  to  be  given  when  their  positions  in  ref- 
erence to  each  other  are  known  or  assumed. 

2.  A  System  of  Coordinates  is  a  system  of  known 
lines,  or  of  lines  and  angles,  for  determining  the  position 
of  points.  The  lines  constituting  a  system  of  coordinates 
are  usually  straight,  but  curved  lines  may  be  used. 

3.  A  Point  is  known  when  its  position  in  reference  to 
a  system  of  coordinates  is  known. 

4.  A  Right  Line  is  one  which  may  be  generated  by  a 
point  moving  in  one  direction  only. 

5.  Illiistrations. — Let  the  position  of  the  lines  OX  and 
0  F  in  reference  to  each  other,  be  assumed,  so  y 

that  the  position  of  a  point  may  be  determined  / 

in  reference  to  them  ;  then  will  they  constitute         y- -^- 

a  system  of  coordinates.     The  known  angle        /         / 

FOXwe  will  constantly  represent  by  gl>.  Tode-     cr ^ X 

termine  the  position  of  any  pointy,  draw  a  line  fio.  i. 

jKi  through  p  parallel  to  the  line  0  F;  and  ph  parallel  to  OX. 
Then  if  Oa  and  Oh  are  given,  the  point  p  will  be  determined- 
For  by  reversing  the  above  process,  we  lay  off  Oa  a  known 
distance  on  OX,  and  draw  ap  parallel  to  0^ ;  then  lay  off  the 
1  1 


FUNDAMENTAL  PRINCIPLES. 


[6 


known  distance  06  on  OY  and  draw  hp  parallel  to  OX. 
Since  the  point  will  be  in  both  the  lines  ap  and  hp,  it  must 
be  at  their  intersection  p. 

The  method  just  described  is  the  one  commonly  used,  but 
the  point  p  may  be  determined  in  other  ways 
in  reference  to  the  same  lines.  Thus,  if  we 
know  the  lengths  of  the  perpendiculars  pc 
andjx?,  the  point  becomes  known.  For,  the 
point  will  be  in  the  line  ep  drawn  parallel  to 
0  Y  and  at  a  distance  from  it  equal  to  the 
perpendicular  Oe  =  pd. 
Similarly,  it  will  be  in  the  line  pf  drawn  parallel  to  OX 
and  at  a  distance  from  it  equal  to  the  per- 
pendicular pc ;  hence  at  its  intersection  with 
ep  at  p. 

Or  the  perpendicular  pc  to  OX  and  the 
parallel  ph  to  the  same  line  may  be  given. 
Or  if  the  perpendicular  dp  be  prolonged 
to  OX,  and^c  to  OY,  then  will  the  lines pg^ 
and  ph  determine  the  point. 

Or  the  lines  pc  and  pd  may  make  any 
known  angles  with  the  lines  OX  and  OY. 
The  system  which  is  adopted  at  the  be- 
ginning of  an  investigation,  must  be   re- 
PiG.  4.  tained  throughout  the  same. 

6.  The  Coordinate  Axes. — When  the  coordinates  con- 
sist of  right  lines  intersecting  at  a  point,  the  right  lines  thus 
used  are  called  coordinate  aoces.  The  line 
OX  is  called  the  axis  of  x,  and  0  Y,  the 
axis  of  y.  These  lines  may  be  prolonged 
indefinitely  in  opposite  directions  from  O, 
dividing  the  plane  which  contains  them 
,  into  four  parts  or  angles. 

Fig.  5.  The  intersection  of  the  axes  at  0,  is 

called  the  origin  of  coordinates. 

Distances  to  the  right  of  the  axis  of  y  and  parallel  to  the 
axis  of  X  are  caWedi  positive  ;  those  to  the  left,  negative  ;  simi- 
larly, those  parallel  to  YY  and  above  XX  are  positive,  and 
those  below,  negaiive. 


+x 


7,8.] 


RECTILINEAR  SYSTEM  OF  COORDINATES. 


The  angle  YOX,  marked  1,  is  called  the  jirst  angle ;  that 
marked  2,  the  second  ;  3,  the  third  ;  and  4,  the  fourth. 

The  rules  in  regard  to  the  signs  and  angles  are  arbitrary, 
and  may  be  changed  when  desired. 

7.  Kinds  of  Coordinates. — If  the  system  consists  of  two 
or  three  lines  meeting  at  a  point,  it  is  called  rectilinear.*  If 
the  axes  form  right  angles  with  each  other,  the  system 
is  called  rectangular.  If  two  axes  only  are 
used,  it  is  called  bilinear.  If  the  axes  make 
oblique  angles  with  each  other,  the  system 
is  called  oblique. 

If  the  system  consists  of  a  fixed  line  OX, 
a  variable  angle  pOX,  and  a  variable  dis- 
tance Op,  it  is  called  a  polar  system. 


Fig.  6. 


Pig.  7. 


The  rectilinear  and  polar  systems  are  tlie  ones  most  commonly  used, 
though  there  are  many  other  systems,  some  of  which 
are  used  occasionally. 

When  the  system  consists  of  three  lines  perpen- 
dicular respectively  to  the  three  sides  of  a  known 
triangle,  it  is  called  trUinear.  This  system  forms  an 
important  part  of  Modern  Geometry. 

The  tangential  system  is  in  a  certain  sense  the  recip- 
rocal of  the  rectilinear.     It  begins  with  a  line,  and,  by  the  intersection  of 
an  infinite  number  of  right  lines,  determines  a  point. 
(See  Salmon's  Conic  Sections.)    It  is  also  used  in  Modem 
Geometry. 

The  system  may  consist  of  a  tangent  to  a  given  curve, 
and  a  radius  from  the  center  to  the  point  of  tangency. 
This  system  is  useful  for  investigating  certain  prop- 
erties of  the  involute. 

The  system  may  also  consist  of  the  arc  of  the  curve 
and  the  angle  which  the  variable  tangent  makes  with 
a  fixed  line.  This  is  caUed  the  intrinsic  equation  of 
the  curve. 

Rectilinear  System  of  Coordinates. 

8.  The  Ordinate  to  a  point  is  the  distance  from  the  point 
to  the  axis  of  x,  measured  on  a  line  parallel  to  the  axis  of  y. 
Thus,  the  ordinate  of^j  is  the  length  of  the  line^iOi,  drawn 

*  Sometimes  it  is  called  the  Cartesian  system,  from  the  latinized  form 
of  Descartes'  name,  who  first  devised  the  system. 


4  FUNDAMENTAL  PRINCIPLES.  [9-11. 

from  pi  parallel  to  OZ  and  terminated  by  tlie  axis  of  a*. 
^Y  Ordinates    are    generally  repre- 

sented by  the  letter  y,  that  letter 
representing   any   ordinate,  and 
"'ir'i       when    a  particular    ordinate   is 
/    «<t    known,  y  may  be  placed  equal 
*/    /      to  that  value.     Thus,  the  ordi- 
f  I  /       nate  of  pi  is   y  =  +  ai^i  =  Ohi ;  of 

'  h    P2y  y^  +  «2P2  =  Oh^ ;  of  ^8,  y  = 

-  (hPs=  -  Oh ;  oipi,y  =  -  aipi= 
*'"■  ^"  ~  Ohi.    Assumed  ordinates   are 

also  designated  by  accents,  or  subscripts,  thus,  y'y  y",  etc. ; 
Vu  y2,  etc. 

9.  The  Abscissa  to  a  point  is  the  distance  from  the  point 
to  the  axis  of  y  measured  on  a  line  parallel  to  the  axis  of  x. 
The  general  value  of  the  abscissa  is  represented  by  the  let- 
ter X.  Hence,  for  the  abscissa  of  pi,  we  have,  x  =  +  b^pi  — 
+  Ooi ;  oip2,  x=  —  bzPi  -■=  —  Oa^ ;  oip^,  x=  —  hps  =  —  Oa^ ; 
of  JO4,  X  =  +&4^4  =  +  Otti.  We  also  use  x',  x'\  etc.,  for  known 
abscissas. 

10.  The  Coordinates  to  a  Point  include  both  the  ab- 
scissa and  ordinate  of  the  point.  A  point  is  indicated  thus : 
the  point  xy,  or  the  point  ab,  or  the  point  (x,  y),  or  the  point 
(3,  —  4) ;  in  which  the  value  of  the  abscissa  is  placed  first 
in  order.  If  the  angle  between  the  axes  is  not  given  it  is 
supposed  to  be  right. 

11.  The  Equations  to  a  Point  are  equations  which  ex- 
press the  values  of  x  and  y  in  terms  of  the  known  abscissa  and 
ordinate.  Thus,  if  a  be  a  known  abscissa,  and  b  a  known 
ordinate,  we  have  for  the  equations  of  the  point  in  the 

1st  angle.  2d  angle.  3d  angle.  4th  angle. 

x=  +  a)  x=  —  a)  X  =  —  a)  x  =  +  a) 

y=+b\'  y=+b\'  y=-b\'  y  =  -  b]  ' 

EXAMPLES. 

1,  If  the  axes  are  rectangular,  locate  the  points  (5,  7);  (—  3,  —4); 
<-4,  3). 

2.  If  ca  =  60°,  locate  the  points  (4,  4) ;  (0,  8) ;  (-  4,  -3). 


13-lG.] 


POLAR  SYSTEM  OF  COORDINATES. 


3.  If  the  axes  are  rectangular,    locate   the   points  (0,  —  3)  •,  (0,  0) ; 
(-2,  0);  (-0,  2). 

4.  If  cj  =  135',  locate  the  points  (3,  2);  (3,  -  2);  (-3,  -  2). 


Fio.  10. 


Polar  Systein  of  Coordinates. 

12.  The  Initial  Line  is  the  line  from  which  the  variable 
angle  is  reckoned.    Thus  if  ^ OX  constitutes 
a  system  of  polar  cor»rdinates  in  which  the 
variable  angle  is  measured  from  the  fixed 
line  OX,  then  will  OX  be  the  initial  line. 

13.  The  Pole  is  the  point  about  which 
the  line  Op  is  conceived  to  revolve.  Thus 
0,  in  the  figure,  is  the  pole. 

14.  The  Radius  Vector  is  the  line  of  variable  length  ex- 
tending from  the  pole  to  the  required  point.  Thus,  if  p  is 
the  point  which  is  to  be  located,  the  line  Op  will  be  the 
radius  vector.  The  length  of  the  radius  vector  we  shall 
generally  represent  by  p. 

16.  The  angle  between  the  radius  vector  and  the  initial 
line  is  called  the  variable  or  vectorial  angle.  We  will 
here  represent  it  by  qj. 

16.  Signs  of  the  Coordinates. — The  angle  cp  is  con- 
sidered ^st'^ive  when  the  radius  vector  revolves  left-handed; 
that  is,  in  a  direction  opposite  to  that  of  the  hands  of  a 
watch ;  and  negative,  in  the  opposite  direction.  Thus,  the 
angle  (or  arc)  ab  is  positive,  and  cd,  negative. 

The  radius  vector  is  considered  positive  when  the  point 
and  the  extremity  of  the  measuring  arc  are  in  the  same 
direction  from  the  pole,  and  negative 
when  they  are  in  opposite  directions  from 
the  pole.  Thus,  in  determining  the  point 
Pi,  a  ab  he  the  measuring  arc,  Opi  will 
be  positive,  for  b  and  pi  are  in  the  same 
direction  from  0  ;  but  if  cdi  be  the  mea- 
suring arc,  the  radius  vector  Opt  will  be 
negative,  for  i  and  pi  are  in  opposite  directions  from  the 
pole. 


Fig.  11. 


6  FUNDAMENTAL   PRINCIPLES.  [17-19. 

17.  The  Polar  Coordinates  of  a  Point  involve  the 
position  of  tlie  fixed  line,  the  vectorial  angle,  and  the  radius 
vector.     A  point  is  indicated  thus,  {qt,  p). 

18.  The  Polar  Equations  of  a  point  consist  in  placing 
q}  and  p  equal  to  determinate  values.  Thus  the  equations  of 
the  point  pi  are 

icp=+abl{rp=+efgl  j  cp  =  -  cdi  ) 

\p=+Op,]'  \p=-0pj'''''       I  p=-0pj' 

and  of  the  point  ^s, 

j  <p  =  +  e/gr   )  \cp=  +ab    )  jcp=-cdi  ) 

]p=+0^3)'  Ip=-0i93r  lp=+Op,\' 

The  points  p^  and  pi  are  indicated  in  a  similar  manner. 

EXAMPLES. 

1.  Construct  the  points' (60°,  5);  (-  45°,  —  3);  (it,  4). 

2.  Construct  the  points  (0°,  0);  (5;r,  —4);    (0°,  —3). 
8.  Construct  tlie  points  {^it,  1) ;  (iff,  —  1) ;  (|ff,  1). 

4.  Construct  the  points  (27r,  2) ;  (27r,  —  2) ;  (0,  2). 

5.  Construct  tlie  points  (0°,  cos  30") ;  (tt,  sin  45°) ;  (—  iff,  tan  45°). 

6.  Construct  the  points  (sin~'  i,  sec  45°);  (ff,  ff);  ('cos-'(— /y/f  ),  _ 

tossin~*jY 

Distance  Between  Two  Points. 

19.  Let  the  Points  be  given  by  their  Rectilinear 
/  Y  Coordinates,  those  of  P  being  x'y',  and  of 

/  p    P\x"y". 

/       ^^^^  I^^t  ^  —  PP  =  ^^®  distance  between  the 

/    ^  ff^——j  g    points,  and  we  have 

/     /       /  ^        f  =  {PBy  +  {BPy-  2PB .  BP'  cos  PBP' 

°  Fig.  12.  ^'  =  {PBf  +  {BPJ  +  2PB  .  BP'  COS  YOX 

=^{0A'-0AY  +  {A'P'-A'By  +  2{0A'-0A).(A'P'-A'B)  cos  co 
=  {x"—x'y  +  {y"—y'y  +  2{x"—x')  {y"—y')  cos  ooy 
which  is  the  formula  required- 


20,  21.]  POLAR  SYSTEM  OF  COOBDHfATES.  J 

20.  Let  the  System  be  Rectangular.— Then  co  =  90°, 
and  tlie  preceding  formula  becomes 

j^=(x"-xr  +  (y"-yr'  (1) 

If  one  of  tlie  points  be  at  tbe  origin,  make  x'=0  and 
y'=  0,  and  we  have  after  dropping  the  accents 

F  =  x'  +  f,  (2) 

which  is  the  square  of  the  distance  of  any  point  from  the 
origin. 

21.  The  Points  being  given  by  Polar  Coordinates, 
to  find  the  distance  between  them. 

Let 

OP'=p\        P'OX=cp'; 
OP  =  p",       POX=cp"', 

then  ' 

POP'  =  <p"  -  cp'. 
From  Trigonometry  we  have 

{PP'f  =  {OPy  +  {OPf  -  WP.OP'  cos  POP'-y 
hence, 

P  =  p'2  +  p"2  _  2p>"  cos  {cp"  -  cp'). 

If  q)"  —  q)'  =  90°,  the  angle  between  the  lines  will  be  right, 
and  the  formula  becomes 

P  =  p'2  +  p'\ 

EXAMPLES. 

1.  Find  the  distance  between  (—  3,  0)  and  (3,  —  5). 

2.  Find  the  lengths  of  the  three  sides  of  a  triangle  whose  vertices  are 
(2,  4),  (0,  0),  and  (-3,  5). 

3.  Find  the  coordinates  of  the  middle  point  of  the  line  whose  ex- 
tremities are  (5,  —1),  and  (—1,  5). 

4.  If  00  =  60°,  construct  the  triangle  whose  vertices  are  (3,  2),  (—  2, 
6),  (1,  -5). 

5.  Find  the  distance  between  the  points  (0°,  2),  and  (—iff,  3). 

6.  Find  the  distance  between  the  points  {\ic,  3),  and  (a",  4). 


CHAPTEE   n. 

THE  RIGHT  LINE  IN  A  PLANE. 
Definitions. 

22.  A  Locus  is  a  series  of  positions  to  which  a  moving 
point  is  restricted  by  some  law.  If  the  successive  positions 
are  consecutive,  the  locus  is  a  line,  either  straight  or  curved. 
If  any  position  is  isolated,  that  part  of  the  locus  is  a  point 
only.  The  locus  of  a  line  may  be  a  surface.  Geometrical 
loci  are  real  points,  lines,  and  surfaces.  Analytical  loci  are 
such  as  are  expressed  algebraically,  and,  in  an  extended 
sense,  not  only  represent  r&d  loci,  but  also  imaginary  points, 
lines,  and  surfaces  and  loci  at  infinity. 

23.  Consecutive  Points  are  such  that  the  distance  be- 
tween them  is  less  than  any  assignable  quantity.  Hence, 
when  compared  with  finite  distances,  the  distance  between  con- 
secutive points  may  be  considered  as  zero,  and  in  such  cases 
two  consecutive  points  may  be  treated  as  one  point.  The 
ratio  between  the  distances  of  different  consecutive  points 
may  be  any  finite  value. 

[In  more  famUiar  language,  we  may  say  that  tlie  distance  between  con- 
secutive points  is  so  exceedingly  small  that  when  added  to  finite  or  measur- 
able- distances,  the  former  may  be  omitted  ■without  appreciable  error.  Thus, 
20,000  miles  4-  tdVo  of  an  inch  and  20,000  miles  +  tooso  of  an  inch  are  pi'ac- 
tically  the  same.  Their  sum  is  practically  40,000  miles,  their  ratio  practi- 
cally 1,  and  their  difference  practically  zero.  Hence  in  regard  to  the  larger 
numbers,  the  small  fractions  may  be  omitted,  and  the  smaller  the  fractions 
the  more  nearly  will  the  larger  part  be  the  true 'value  of  the  expression. 
When  the  fraction  is  less  than  any  assignable  quantity,  it  cannot  be  mea- 
sured or  expressed,  but  the  larger  part  is  measurable,  and  is  called  finite. 
These  remarks  are  true  whatever  be  the  actual  value  of  the  small  fractions. 
Thus,  in  the  preceding  example,  the  ratio  of  the  first  fraction  to  the  second 

8 


24^27.]  EQUATION  OF  A  RiaHT  LINE.  9 

is  10.  If  the  fractions  had  been  jiyuTJo  and  nnrjoo)  their  ratio  would  have 
been  25  ;  while  the  sum  of  the  entire  quantities  would  have  been  practically 
40,000,  their  difference  zero,  and  their  ratio  1.  Those  fractions,  however 
small,  may  have  any  conceivable  ratio.  Quantities  which  are  less  than 
any  assignable  quantity  are  called  ivfinitesimals.  One  object  of  the  Differ- 
ential Calculus  is  to  find  the  ratio  of  infinitesimals  when  related  to  each 
other  by  known  laws] 

24.  The  Equation  of  a  Locus  is  an  equation  wliicli  ex- 
presses tlie  relation  between  the  coi'trdinates  of  every  point  of 
tlie  locus.  The  distance  from  the  origin  to  the  point  where 
the  locus  cuts  an  axis  is  called  the  Intercept  on  that  axis. 

25.  Analytical  Geometry  consists  in  expressing  geo- 
metrical quantities  by  algebraic  symbols ;  establishing  an 
equation  of  the  locus  by  means  of  these  symbols,  then  sub- 
jecting the  quantities  and  the  equation  to  algebraic  opera- 
tions so  as  to  deduce  the  properties  of  the  locus,  and  finally 
interpreting  the  result. 

26.  Coordinate  Geometry  is  that  system  of  geometry  in 
which  the  points,  lines,  or  surfaces  which  are  to  be  considered 
are  referred  to  a  system  of  coordinates,  and  their  properties 
determined  by  means  of  their  relations  to  those  coordinates. 

Equation  of  a  Right  Liiie. 

27.  To  find  the  Rectilinear  Equation  to  a  Right 
Line. — Let  BP  be  the  line  cutting  the 

axis   of  ic  at   5  and  the  axis   y  aX  F.  I 

Take  any  point  P  on  the  line  and  draw  /    ^ 

PD  parallel  to  OY;  then  will  /X^ 

x  =  OD,        y  =  DP.                     ^y^"T    , 
Let  e=PBD,     h^OF.  -y^ J         ^ 

Through  F  draw  FG  parallel  to  OB,  ^'°  ^^■ 

then  the  triangle  PFO  gives 

FG'.PG  '.'.  sin  FPG  :  sin  {PFG  =  FBX), 

or  X  :y  —  b  :  :  sin  (ca  —  ^)  :  sin  d 


.•.y  = 


sm 


sin   (a?  —  B) 


x  +  b.  .  (1) 


10 


THE  RIOET  LINE. 


[28,  39. 


If  wi'  be  substituted  for  the  coefficient  of  x  tlie  equation 
becomes 

y  =  m'x  +  b;  (2) 

which  is  the  equation  sought. 

28.  Rectangular  Equation  to  a  Right  Line. — Make 
y  CO  —  90°  in  the  equation  (1)  of  the  preceding 

Article,  and  we  have 


y 


sin  6 
cos  6 


x  +  b; 


Fig.  15. 


or 


(3) 
(4) 


y  =  tan  d.x  +  b. 

Let  m  =  tan  0,  and  we  have 

y  =  mx  +  b; 

which  is  the  required  equation. 

29.  Equation  to  a  Right  Line  in  Terms  of  its  In- 
tercepts.—Equations  (2)  and  (4)  are  in  terms  of  the  inter- 
cept (b)  on  the  axis  of  y  and  the  inclination  of  the  line,  which 
is  the  more  common  form.  But  it  may  be  found  in  terms  of 
the  intercept  on  both  axes. 

Let  a  =  OB  be  the  intercept  on  the  axis  of  x,  OF  =  6, 
the  intercept  on  the  axis  of  y. 

In  Figs.  14  and  15  we  have 


OF 
OB 


or 


sm  OBF 

sm  OFB         *  ' 

b 
=  m  or  m, 


where  a  is  negative  because  B  is  at  the  left  of  the  origin. 
Substituting  the  value  of  m'  in  equation  (2),  and  m  in  (4) 
gives 


a^b      ■^* 


(5) 


which  is  the  required  equation. 


30,31.]  EQUATION  OF  A  RIGHT  LINE.  IX 

30.  Equation  to  a  Right  Line  in  terms  of  the  Per- 
pendicular upon  it  from  the  Origin  and  the  Angle 
made  by  the  Perpendicular  with  the  Axis  of  x. 

Let  p  =  OR  be  tlie  perpendicular  from  the  origin  upon 
the  line   CD ;   a  =  OD,  the  intercept  on 
the  axis  oi  x;  h  =■  OC,  the  intercept  on 
the  axis  of  y.     Then 


a  =  — ^— ,         0  =  -^ — 
cos  a  sm  a 


B 
Fig.  16. 


and  these  values  substituted  in  equation  (5)  give 

X  cos  (X  +  y  sin  a  =  p,  (6) 

which  is  the  required  equation. 

If  the  axes  are  oblique,  the  equation  becomes 

X  cos  a  +  y  cos  {go  —  a)  =  p.  (7) 

31.  Of  Constants  and  Variables. — A  constant  is  a 
quantity  which  retains  the  same  value  during  a  particular 
discussion.  Constants  are  of  two  kinds-— ft  .fed  and  arhitrary. 
Fixed  constants  are  numerals,  as  3,  7,  12,  etc.  Arhitrary 
constants  are  those  which  admit  of  different  values.  Thus 
in  equation  (4)  m  may  have  any  value  assigned  to  it,  but 
when  once  assigned  it  becomes  ^^a-ec?  for  that  particular  dis- 
cussion ;  and  the  same  is  true  of  h.  Arbitrary  constants  are 
usually  represented  by  the  first  letters  of  the  alphabet.* 

Variables  are  quantities  which  may  have  an  unlimited 
number  of  values  in  a  particular  discussion.     Thus,  in  equa- 

*  In  higher  analysis  arbitrary  constants  are  made  to  vary  ;  or,  in  other 
words,  certain  quantities  which  are  usually  considered  constant  are  made  to 
vary.  Thus,  in  the  equation  of  a  right  line,  y  =  mx  +  b,  if  &  varies  while 
m  remains  fixed,  the  line  will  generate  a  plane. 

Similarly,  the  parameter  of  a  parabola  may  increase  at  a  uniform  rate 
while  points  which  generate  parabolas  shoot  out  constantly  from  the  vertex 
with  an  infinite  velocity.  Chirves  thus  generated  are  called  "A  family  of 
curves." 

An  arbitrary  constant  may  be  defined  as  a  quantity  wJiich  varies  infinitely 
slow  comvared  with  the  generator  of  the  curve. 


12  THE  RIOET  LIKE.  [33-34 

tion  (4),  (5),  or  (6),  x  may  have  all  conceivable  values  from 
+  00  to  —  00,  and  the  corresponding  value  of  y  may  be 
determined  from  the  equation ;  hence,  y  may  also  have  an 
unlimited  number  of  values.  The  real  values  of  the  variables 
in  an  equation  are  often  confined  within  finite  limits,  as  we 
shall  see  in  the  case  of  the  circle,  ellipse,  etc. ;  still,  the 
number  of  real  values  within  those  limits  is  unlimited. 

32.  The  Absolute  Term  is  that  term  of  an  equation 
which  contains  no  variables.  The  quantity  h  in  the  equation 
of  the  right  line  is  the  absolute  term.  If  an  equation  has  no 
absolute  term,  the  locus  passes  through  the  origin  ;  for  the  inter- 
cepts for  that  point  are  both  zero. 

33.  The  Discussion  of  an  Equation  to  a  Locus 

consists  in  interpreting  the  results  deduced  from  the  equa- 
tion. The  general  process  consists  in  solving  the  equation 
in  reference  to  one  of  the  variables  and  assuming  fixed 
values  for  the  other,  and  interpreting  the  results. 

34.  Discussion  of  the  Rectangular  Equation  to  the 
Right  Line. — The  equation  is  (Art.  28), 

y  =  mx  +  b.  (1) 

1°.  Let  X  =  0,  then  y  =  b. 

These  are  the  cofirdinates  of  a  point  on  the  axis  of  y  at 
a  distance  b  from  the  origin.  It  is  the  point  where  the  line 
intersects  the  axis  of  y,  and  the  ordinate  to  this  point  is  the 
Intercept  on  that  axis,  (Art.  24). 

2\  Let  y  =  0,  then  x=  -~=-  — ^  =  -  6  cot  d. 

m  tan  6 

These  values  of  x  and  y  are  the  coordinates  of  the  point 
where  the  line  intersects  the  axis  of  x ;  and  the  value  of  x  is 
the  intercept  on  that  axis. 

3°.  Let  6  =  0,  then  we  have 

y  =  mx,  (2) 

which  is  the  equation  to  a  right  line  passing  through  the 
origin,  (Art.  32). 


34.]  EQUATION  OF  A  RIGHT  LINE.  13 

4°.  Let  ra  =  0,  then  tan  6  =  0  and  0  =  0,  and  we  have 
7/  =  0.x  +  b.  (3) 

But  when  ^  =  0  the  line  is  parallel  to  the  axis  of  x ;  hence 
this  is  the  equation  to  a  right  line  parallel  to  the  axis  of  ./• 
and  at  a  distance  from  it  equal  to  h.  In  this  equation  the 
values  of  x  and  y  are  independent  of  each  other. 

5^.  Let  m  =  0,  and  h  =  0,  then  will  the  line  coincide  with 
the  axis  of  x,  and  the  equation  becomes 

2/  =  0.x,  (4) 

which  is  the  equation  of  the  axis  of  ./'.  In  this  equation  ?/  is 
zero  for  all  values  of  x. 

6°.  Let  m  =  cc;  then  tan  6  =  <x,  and  6  =  90^,  and   the 
equation  becomes 

y  =  CD.x  +  b; 


or 


or 


1      _        A. 

—  y  _a?  +  0D> 

O.y  =  x  +  0. 


(5) 


In  this  equation  x  is  zero  for  all  values  of  y  ;  hence  it  is  the 
equation  to  the  axis  of  y. 

The  equation  to  a  line  parallel  to  the  axis  of  y  is 


O.y  =  X  +a. 

T.  When  m  and  h  are  both  positive,  the 
line  lies  across  the  second  angle.  For,  h  being 
positive,  the  line  cuts  the  axis  of  y  above  the 
origin;  and  m  being  positive  makes  tan  6 
positive,  hence  the  angle  6,  above  the  axis  of  x 
and  at  the  right  of  the  line,  will  be  acute. 

8°.  When  h  is  positive  and  m  negative,  it  lies 
across  the  first  angle. 


(6) 


Fig.  la 


14'  THE  BIGHT  LINE.  [35, 36. 


9".  Wlien  h  is  negative  and  m  positive,  it  lies 


Fig.  19. 
Y 


P  B        across  tlie  fourth  angle. 


V 


X 


10°.  When  h  is  negative  and  m  negative,  the 


p       line  lies  across  the  third  angle. 

Fig.  20. 

[Obs. — It  is  not  necessary  for  the  student  to  remember  all  these  results, 
but  it  is  very  important  that  he  should  be  able  to  make  the  discussion,  and 
interpret  the  results.  Much  practice  in  interpreting  equations  is  advis- 
able.] 

35.  Every  Equation  of  the  First  Degree  between 
Tw^o  Variables  is  the  equation  to  a  right  line. 

The  most  general  form  of  the  equation  of  the  first  degree 
between  two  variables  is 

^a?  +  %  +  0  =:  0 ; 

in  which  A,  B,  C,  are  arbitrary  constants.     Solving  in  refer- 
ence to  y  gives 

A        G 

y  =  -B''-B' 

A  C 

If  —  ^  be  represented  by  m,  and  —  ^  by  &,  the  equation  be- 
comes 

y  =  mx  +  h; 

and  therefore  represents  a  right  line,  (Arts.  27  and  28). 

In  an  algebraic  sense,  the  values  of  x  and  y  in  the  preced- 
ing equation  are  indeterminate,  and  since  this  is  generally 
true  for  the  equations  to  loci,  this  branch  of  mathematics  is 
sometimes  called  Indeterminate  Geometry. 

36.  Polar  Equation  to  a  Right 
Line.— Let  BChe  the  line,  OX  the  initial 
line,  OB  =  p  the  perpendicular  from  the 
pole  O  upon  the  line,  the  angle  BOX=  a, 
P  any  point  of  the  line,  OP  —  p  the  radius 
vector,  and  9?  =  POX  the  variable  angle. 


87.]  POLAR  equation:  16 

The  right-angled  triangle  ORP  gives 
OP  cos  P  OR  =  OB, 

or  p  cos  (</>  —  a)  =  J) ; 

which  is  the  required  equation. 

The  angle  n  is  positive  when  it  is  generated  by  a  left- 
handed  rotation,  or,  more  generally,  from  +  x  towards  +  y. 

p 

Discussion. — 1°.   Let  qj  —  0,  then  p  -■  OB  = ; c-  = 

cos  (—  iy) 

p  sec  tx ;  which  is  the  intercept  on  the  initial  line. 

2°.  Let  (p  =  a,  then  p  =  OR  =  — ^  =  p. 

cos  0       ^ 

3°.  Let  ^  =  90''+  a,  then  p  =  -     ^rr.  =  ^   as  it  should, 
^  '       cos  90 

since  the  radius  vector  becomes  parallel  to  the  line,  and 

hence  cannot  meet  it  at  a  finite  distance. 

4°.  Let  r/^>90°  +  a  and  <180''  +  ^f,  then  will  cos  {qj  —  a) 

be  negative  and  the  radius  vector  p  will  also  be  negative, 

and  the  point  thus  determined  will  be  below  B. 

5°.  Let  cp  =  180°,  then  p  = ^£^-, ^  =  -^=^,  which 

cos  (180  —  a)         cos  a 

gives  the  point  B  again. 

6°.  Let   cp  =  180°  +  a,  then   p  —  ^^g^gQ^  ^  —  V->  which 

gives  the  point  R. 

7°.  Let  fl'  =  0,  then  the  equation  becomes  /o  cos  <p  =  p, 
which  is  the  polar  equation  of  a  line  perpendicular  to  the 
initial  line. 

8°.  Let  OL  —  90°,  then  we  have  /)  sin  <^  =  p  for  the  equa- 
tion to  a  line  parallel  to  the  initial  line. 

9°.  \iq)^7T,  27r,  drr,  etc.,  then  p=  ±pseGa=  OB.  Gen- 
erally, all  the  points  of  the  line  will  be  determined  once 
by  making  <p  pass  from  0°  to  180° ;  and  again  by  passing 
from  180°  to  360°,  and  still  again,  from  2;r  to  37r  and  so  on 
indefinitely. 

37.  To  Trace  a  Right  Line.— Two  points  determine  a 
right  line  ;  therefore,  assume  two  different  values  for  one  of 


16  TEE  BIGHT  LINE.  [87. 

tlie  variables  and  find  the  corresponding  values  for  tlie  other 
by  means  of  the  equation  of  the  line.  Construct  the  points 
thus  found  and  draw  a  right  line  through  them. 

If  the  axes  are  rectilinear,  the  easiest  way  is  to  find  the 
intercepts  and  lay  them  off  on  the  respective  axes.  There- 
fore, make  ?/  =  0  in  the  equation  and  find  x,  and  lay  its 
value  off  on  the  axis  of  x.  Similarly,  make  £c  =  0,  find  y  and 
lay  it  off  on  the  axis  of  y ;  the  line  drawn  through  the  points 
thus  found  will  be  the  line  required. 

If  the  line  is  given  by  its  polar  coordinates,  find  the  inter- 
cept on  the  axis  of  x  by  making  ^  =  0  in  the  equation,  and 
draw  the  line  through  it  and  the  extremity  of  the  perpen- 
dicular from  the  origin. 


EXAMPLES. 

1.  Determine  three  points  in  the  right  line  4y  =  —  6a;  +  3,  whose 
abscissas  are  respectively  —  3,  1,  and  2,  and  draw  the  right  line  through 
the  points  thus  found. 

Making  a;  =  —  2  in  the  equation  of  the  line,  we  find  y  =  3|.  With 
any  scale  of  equal  parts  lay  off  on  —  a;  two  divisions 
of  the  scale  =  OJ?,  and  at  B  draw  BB  parallel  to  the 
axis  of  y  and  lay  off  on  it  +  3f  of  the  same  equal 
parts. 

— I -A  Similarly,  for  x  =  l  we  find  y=  —  \.     For  this 

lyl  point,  lay  off  one  space  =  OB^   to  the  right  of  the 

>Pa  origin,  and  ^2^2  =  4  of  a  space  below  the  axis  of  a-. 

Fig.  22.  which  gives  P^  for  the  required  point. 

For  a;  =  2,  we  find  ?/  =  —  3^,  by  means  of  which  the  point  P3  is 
found. 

If  the  construction  is  correct  the  three  points  will  be  in  a  right  line. 
If  a;  =  0,  y  =  \  which  is  the  y-intercept.      If  y  =  0,  a;  =  ^  which  is 
the  x-intercept. 

2.  Construct  the  line  ^y  =  \x  —  \. 

[Make  a;  =  0,  and  find  y  =  —  \  which  is  the  y-intercept.  Similarly 
the  x-intercept  is  |.] 

3.  What  angle  does  the  line  y  =  x  make  with  the  axes  ?  What  are 
its  intercepts  ? 

4.  Wliat  angle  does  the  line  y  =  —  fa;  —  4  make  with  the  axis  of  a?  ? 
What  are  its  intercepts  ? 

5.  Trace  the  line  iyv^S  =  —  3a;\/|  —  1. 

6.  Trace  the  line  3a!  —  4y  +  2  =  0. 


88.]  THE  POINT  AND  BIGHT  LINE.  17 

7.  What  two  lines  are  represented  by  the  equation  y'  =  4*5  ?  What 
angle  do  they  make  with  each  other  ? 

8.  What  two  lines  are  represented  by  the  equation  a;-  =:  4  —  4^  +  2/*  ? 
At  what  point  do  they  intersect  each  other  ? 

9.  What  are  the  intercepts  for  the  two  lines  represented  by  the  equa- 
tion {x  —  2y  =  2/^  +  6y  +  9  ?  Construct  the  lines  and  find  where  they 
intersect  each  other. 

10.  Trace  the  line  p  cos  (^  —  30')  =  6.     Also  p  sin  9)  =  —  5. 

X      y 

11.  Deduce  the  equation  of  the  line  — +-|^=  1  from  a  fiffure. 

at)  ® 

12.  Deduce  the  equation  x  cos  a  +  y  sin.  a=p  from  a  figure. 

13.  Construct  the  line  whose  equation  is  2/  =  3^  —  5,  when  00  =  60°. 

14.  Interpret  the  equation  p  cos  dcp  =  0. 


The  Point  and  Right  Line. 

38.  Equation  to  a  Right  Line  -which  shall  pass 
through  a  given  point. — Let  the  given  point  be  (x',  y'), 
and  since  it  is  on  the  right  line 

y  =  mx  +  b  (1) 

it  must  satisfy  the  equation  of  that  line,  hence  we  will  have 

y'  =  7nx'+b.  (2) 

Eliminating  h  between  these  equations  gives 

y-y'  =  m{x-x');  (3) 

which  is  the  required  equation.  The  value  of  w  in  this 
equation  is  indeterminate,  and  an  indefinite  number  of 
values  may  be  assigned  to  it ;  hence  the  equation  is  an  ex- 
pression for  the  well-known  fact  that  an  infinite  number  0/ 
right  lines  may  be  passed  through  a  given  point. 
From  (2)  we  have 

y'-b 

X 

which  substituted  in  (1)  gives 

y  =  ^-^x  +  b;  (4) 


18  THE  BIGHT  LINE.  [39, 40. 

whicli  is  another  form  of  tlie  equation,  but  not  being  so  con- 
venient as  (3)  it  is  rarely  used  in  practice. 

39.  Equation  of  Condition, — Observe  that  equation 
(2)  of  the  preceding  article  is  not  the  equation  of  a  line,  for 
it  contains  no  variables ;  neither  is  it  the  equation  of  a 
point.  It  is  a  true  equation  between  constants,  by  means  of 
which  the  values  of  the  arbitrary  constants  may  be  deter- 
mined. The  equation  is  not  only  true,  but  is  established 
upon  certain  conditions;  the  conditions  being  determined  by 
the  geometric  forms,  or  by  the  equations  which  represent 
those  forms.  Such  equations  are  called  Equations  of  Condi- 
tion. 

40,  The  Equation  to  a  Right  Line  passing  through 
Two  given  points. — Let  {x,  y'),  {x",  y")  be  the  two  points 
through  which  the  right  line 

y  =  mx  +  h  (1) 

must  be  made  to  pass.  The  coordinates  of  the  points  must 
satisfy  the  equation  to  the  line ;  hence  we  have  the  two 
equations  of  condition 

y'  =  mx'  +  h,  C2) 

y"  =  mx"  +  b,  (3) 

by  means  of  which  the  arbitrary  constants  m  and  b  may  be 
completely  determined  in  terms  of  x'y'  and  x'y".  Subtract- 
ing (2)  from  (3),  we  have 

y"  -y'  =  m  (x"  -  x) ; 

,'.m  =  K;^.  (4) 

X  —x  ^  ' 

Subtracting  (2)  from  (1),  and  substituting  the  value  of  w  in 
the  result,  gives 

y-y'  =  |^|(x-a.');  (6) 


40.]  EXAMPLES.  19 

which  is  the  required  equation.     It  may  be  put  under  the 
form 

y  -  y'  _y"  -  y' 


X  —  X         X    —  X 


(6) 


If  one  of  the  points,  as  (x",  y")  be  at  the  origin,  then  x"  =  0, 
y"  =  0,  and  the  equation  becomes 

y  =  -tx  (7) 

which  is  the  equation  of  a  right  line  passing  through  the 
origin,  and  making  an  angle  with  the  axis  of  x  whose  tan- 

i  •    y 

gent  18  ^,. 

If  one  of  the  points  is  (0,  b)  we  have 

v'  —  h 
y  =  - — ; —  x  +b  =  mx  +  h ; 

which  is  the  ordinary  equation  to  the  right  line. 


EXAMPLES. 

1.  Find  the  equation  to  a  right  line  which  makes  an  angle  of  45° 
with  the  axis  of  a*,  and  passes  through  the  point  (2,  3). 

Here  m  =  tang  45°  =  1,  and  equation  (3)  Art.  38  becomes 
y  =  x-\-l, 

which  is  the  required  equation. 

2.  Find  the  equation  to  a  riglit  line  which  passes  through  the  point 
(—  2,  3),  and  whose  intercept  on  y  is  1 ;  and  find  the  angle  wliich  the 
line  makes  with  the  axis  of  x. 

(Use  Eq.  (4)  Art.  38,  or  Eq.  (o)  Art.  40.) 

3.  Find  the  equation  to  a  right  line  which  makes  an  angle  of  150° 
with  the  axis  of  x  and  passes  through  the  point  (—  4,  —  5). 

4.  Find  the  equation  of  the  right  line  which  passes  through  the 
points  (3,  -  4)  and  (-  2,  6),  and  find  tan  6.     (Eq.  (5),  Art.  40.) 

5.  Form  the  equations  of  the  sides  of  the  triangle  the  coordinates  of 
whose  vertices  are  (—  3,  0),  (0,  —  4),  (1,  0). 

6.  Find  the  equation  of  a  right  line  passing  through  the  points  (0, 
0),  (-3,2). 


THE  RIGHT  LINE.  [41.  43. 

Relations  between  Two  Lines. 

41.  To   find   the   Point   of  Intersection   of    Two 
Lines. — Let  the  equation  of  the  line  £C 

Y  f/      be 

U^      and  of  EF, 

y      \  y  =  w'ic  +  b'. 

X        I    \    V    If  the  lines  intersect  there  will  be  one 
B  /e    °         point  in  common,  and  hence  the  coiirdi- 

Fio.23.  nates  for  that  point  must  satisfy  the  equa- 

tions to  both  lines.  To  find  the  values  of  x  and  y  which 
will  satisfy  this  condition,  we  have  only  to  consider  the 
preceding  equations  as  simultaneous,  and  eliminate  x  and  y 
successively  by  the  well  known  rules  of  algebra.  Letting 
the  resulting  values  of  x  and  y  be  denoted  by  x^  and  yi,  we 
have 

h'  —  h       ^  ^  mV  —  m'h      -r.  n 

Xi  = ,  =  OB ;  Wi  = —  =  DP. 

m  —  m  m  —  m 

lim  =  m'  we  have 

Xi  =  GO,  and  1/1  =  cd; 

hence  the  lines  will  not  intersect  at  a  finite  distance,  or,  in 
other  words,  they  will  be  parallel 
The  equation 

m  =  m' 

is  the  equation  of  condition  of  parallelism  of  two  right  lines. 
Jim  =  m'  and  b  =  b',  we  have 

0  0       ' 

both  of  which  are  indeterminate,  hence  the  lines  will  coincide 
throughout. 

42.  Equation  to  a  Line  -which  is  Parallel  to  a  given 
Line. — Let  the  given  line  be 

y  =  m'x  +  b'.  (1) 

The  form  of  the  equation  of  the  required  line  will  be 

y  =  mx  +  b;  (2) 


43,44.] 


RELATIONS  BETWEEN  TWO  LINES. 


21 


but,  according  to  tlie  equation  of  condition,  m  =  m,  whicli 
value  in  equation  (2)  gives 


y  =  m'x  +  &, 


(3) 


for  the  required  equation.  In  this  equation  the  arbitrary 
constant  h  is  undetermined;  hence  the  equation  expresses 
the  well-known  fact  that  an  infinite  number  of  lines  may  he 
dravm  parallel  to  a  given  line. 

43.  Equation  to  a  Right  Line  passing  through  a 
given  Point  and  Parallel  to  a  given  Line. — Let  the  given 
point  be  {x',  y').  Let  equation  (1)  of  the  preceding  article  be 
the  given  line,  then  will  (3)  be  parallel  to  it,  and  it  remains 
to  make  (3)  pass  through  the  given  point.  Hence  the  coordi- 
nates of  the  point  must  satisfy  equation  (3),  and  we  have  the 
equation  of  condition 

y  —  m'x  +  h ; 
which,  being  subtracted  from  equation  (3),  gives 

y  —  y'  =  m'  (x  —  x'), 
which  is  the  required  equation. 

44.  Angle  between  Two  Lines.— Let  the  equations  be 


y  =  mx  +  h,  (for  BF), 
y  =  m'x  +  b',  (for  EF). 

From  the  figure  we  have  6'  equal  to 
the  sum  of  6  and  0,  or 


^  =  6'- 


.'.  tan  /?  = 


Y 

/ 

'/• 

/ 

0 

/ 

d 

/b 

E 

Fig.  34. 


tan  6'  —  tan  6 
1  +  tan  6  tan  6' 


m 


m 


i  +  mm! ' 

from  which  /?,  the  required  angle,  may  be  found. 

If  m  =  m',  then  yS  =  0,  or  the  lines  are  parallel,  as  pre- 
viously shown. 


22  THE  RIOHT  LINE.  [45, 46. 

If  the  lines  are  perpendicular  to  eacli  other,  yS  =  90°,  and 
tan  yS  =  00,  hence  we  must  have 

1  +  mm  =  0, 

which  is  the  equation  of  condition  that  two  right  lines  shall  be 
mvtuaMy  perpendicular. 

45.  Equation  to  a  Right  Line  perpendicular  to  a 
given  one. — Let  the  given  line  be 

y  =  m'x  +  b', 

and  let  it  be  required  to  make  the  line 

y  =  mx  +  b 

perpendicular  to  it.     The  equation  of  condition  gives 

1 

m= r, 

m 

in  which  m'  is  known  from  the  given  equation,  and  hence  m 
becomes  known ;  and  by  substituting  its  value  in  the  pre- 
ceding equation,  we  have 

y= ix  +  b, 

which  is  the  required  equation. 

Since  the  arbitrary  constant  b  remains  undetermined,  the 
equation  expresses  the  well-known  fact,  that  an  infinite  num- 
ber of  lines  may  be  drawn  perpendicular  to  a  given  line. 

46.  Equation  to  a  Right  Line  perpendicular  to  a 
given  line  and  passing  through  a  given  point. — Let 
the  point  be  {x,  y') ;  and  the  given  line  be 

y  —  m'x  +  b ; 

then  will  the  required  line  be  of  the  form,  (Art.  45), 

1 

y  = -,x  -\-  b. 

Since  this  line  is  to  pass  through  the  point  we  have  the  equa- 
tion of  condition 

y  =  — ix  +6, 


47.]  EXAMPLES.  23 

wMcli  subtracted  from  tlie  preceding  equation,  gives 

2/  -  2/'  =  -  —'  (a^  -  x'), 

which  is  the  required  equation. 

47.  Equation  to  a  Right  Line  passing  through  a 
given  point  and  cutting  a  given  line  at  a  given 
angle. — Let  the  given  line  be 

y  =  m'x  +  h ; 
and  the  required  line  be 

y  —  y'  =  m(x  —  x') ; 

and  ^  the  given  angle.     Then,  according  to  Article  44,  we 
find 

m  —  tan  6 
1  +  m  tan  p 

which  substituted  in  the  preceding  equation  gives 

m'  —  tan  /3    ,  ,. 

^      ^       1  +  m  tan  /i  ^  ^ 

for  the  required  equation.     If  /5  =  0  the  equation  becomes 

y  —  y'  =  m  {x  —  x'). 

li  /3  =z  90°,  the  equation  becomes,  by  dividing  the  numerator 
and  denominator  by  tan  /3, 

y-y  =  -:^'  (^— ^)- 

EXAMPLES. 

1.  Find  the  coordinates  of  the  point  of  intersection  of  the  lines 

Sx-4p  =  l;  y-2x=S. 

2.  Find  the  coordinates  of  the  point  of  intersection  of  the  lines 

2      3'-    '  5      7~ 


24  THE  BIGHT  LINE.  [47. 

3.  Find  the  angle  between  the  lines 

y  =■  A:X+  o\  2y  =  Qx  —  T. 

4.  Find  the  angle  between  the  lines 

dy-4x=i0;  ?/  =  7,t  -  ^. 

5.  Find  the  equation  of  a  right  line  perpendicular  to  the  line 
y  =  7a;  —  4. 

6.  Find  the  equation  of  a  right  line  perpendicular  to  the  line  y  =x, 
and  passing  througli  the  point  (1,  1)  in  the  line. 

7.  Find  the  equation  of  a  right  line  perpendicular  to  the  line  2y  — 
4a;  =  7,  and  passing  through  the  point  (—  3,  —  4). 

8.  Find  the  equation  of  a  right  line  parallel  to  y  =  —  7x  +  A,  and 
passing  through  the  point  (—  2,  3). 

9.  Find  the  perpendicular  distance  between  the  parallel  lines 

y  =  4x  —  S-,  y  =  4x  +  4:. 

10.  Find  the  perpendicular  distance  of  the  line  y  =  —Bx  +  5  from 
the  origin. 

11.  Find  the  length  of  the  pei-pendicular  from  any  point  to  a  right 
line. 

Solution. — Let  the  point  be  {x,  y)  and  the  line  be 

y  =  mx  +  5; 
then  will  the  equation  of  the  perpendicular  be 

y-y'=-  —  {x-^')' 

Eliminating  y  and  x  successively  between  these  equations  gives  the 
coordinates  of  the  point  of  intersection  of  the  line  and  perpendicular. 
Letting  this  point  be  x"y'\  we  have 

,,      my  —  mb  +  x' 

*   ~       l  +  m«       ' 

m^y'  +  mx'  +  & 


*    ~  1  +  m* 

From  these  we  find 

y  —  mx  —  h 


x'  —  X'  =  m 
y"  -y'  = 


1  +OT«     ' 
—  y'  +  mx  +  ft 


l+TO* 

and  the  length  of  the  perpendicular  will  be 

, y'—mx  —i      ^ , 

Vi^"- ">')'+ (3/"- yr  =  .^/jrr^  -  -^(^^y)- 


47.]  EXAMPLES.  25 

If  the  equation  of  the  line  be  of  the  form  Ax  +  By  -\-C  =  (i,  then 

^^Ax'  +  By'+C 
^/A'  +B' 

12.  Find  the  distance  of  the  point  (—  2,  3)  from  the  line  y  =  2x  —  i. 

13.  Find  the  distance  of  the  point  x'y'  from  the  line 

X  cos  a  +  y  cos  {co  —  a)  =p. 

Solution. — The  equation  of  a  parallel  line  passing  through  the  given 
point  will  be 

x'  cos  (X  +  y  cos  (co  -  a)  —  j/, 

and  the  required  distance  will  be  p  —  p.     Subtracting  p  from  the  left 
member,  we  have  for  the  length  of  the  required  jyerj^endicitlar 

±  (x'  cos  a  +  y'  cos  (cj  —  a)  —  ^>), 

in  which  the  plus  sign  is  used  when  tlie  origin  and  the  point  are  on 
opposite  sides  of  the  line,  and  minus  when  they  are  on  the  mme  side. 

If  the   coordinates   are   rectangular,    the  length   will  be,    making 

CO  =  90% 

±  {x'  cos  OL  —  y  sin  a  — p'). 

14.  Given  the  area  (a)  and  base  (Z»)  of  a  triangle  to  find  the  locus  of 
the  vertex. 

Let  the  equation  of  the  base  be 

X  cos  «  +  1/  sin  a  —  p'  =  0 ; 
then  X  cos  a  +  i/'  sin  a  —  ^ 

will  be  the  length  of  the  perpendicular  from  the  vertex  (.c,  y")  upon  the 

base ;  which  is  also  2a  -^  & ;  hence  if  x  and  y  be  general  variables,  we 

have 

2a 
«  cos  a  +  y  sin  a  —  ^  =  J-, 

which  is  the  equation  of  a  right  line, 

15.  To  find  the  equation  to  a  line  which  shall  pass  through  the 
intersection  of  two  given  lines. 

Solution.— T\\Q  point  of  intersection  may  be  found  as  by  Article  41, 
and  the  equation  determined  by  Article  38.  An  equation  may,  however, 
be  written  without  finding  the  point  of  intersection.  Let  the  equa- 
tions be 

Ax->fBy-^C  =  %  and  Ax  +  By+  C  =  Q. 

Multiplying  the  former  by  some  constant  as  m  and  the  latter  by  another 
constant  as  /i,  and  adding  the  results,  we  have 

m{Ax  +  By+G)-^n  {Ax  +  Bfy  +  C)=0 


26  THE  BIGHT  LINE.  [47. 

for  the  required  equation.  For  it  is  the  equation  to  some  line,  being  of 
the  first  degree,  and  for  the  point  of  intersection  of  the  lines  both  terms 
will  be  zero ;  hence  the  new  line  will  pass  through  that  point.  Since  m 
and  n  are  arbitrary  there  may  be  an  infinite  number  of  lines  through  the 
point.  If  n  be  fractional,  all  possible  lines  may  be  represented  when 
m  =  l. 

16.  Find  the  equation  to  the  line  which  passes  through  the  point  of 
intersection  of  the  lines  3a;  —  4y  —  3  =  0,  and  2a;  +  7^/  —  5  =  0. 

17.  Find  the  equation  to  a  right  line  which  bisects  the  angle  between 
the  lines 

Sy  +4:X=  12,  and  5y—2x  =  0. 

Solution. — Let  a;',  y  be  ,any  point  on  the  bisecting  line,  then  will  the 
perpendiculars  from  this  point  on  the  lines  be  equal  to  each  other. 
Using  the  value  of  P  in  the  answer  to  the  11th  example  above,  we  shall 
have  for  the  perpendicular  upon  the  first  line 

4a;'  +  Sy  —  12 
V4*T^ 

and  on  the  second  ~       "*"    j^. 

-v/4  +  25 

Placing  these  equal  to  each  other  and  reducing  gives 

31-52x'  -  8-8&y  -  64-56. 

But  since  x  and  y'  are  the  coordinates  of  any  point  on  the  line  they 
maybe  considered  as  running  variables,  and  the  accents  maybe  omitted. 
Hence  the  equation  of  the  bisecting  line  is 

31-52a;-8-86t/  =  64-56. 

The  line  perpendicular  to  this  line  bisects  the  supplementary  angle. 

18.  Find  the  equations  of  the  lines  bisecting  the  angles  between  the 
lines  whose  equations  are 

12a;  +  5y  =  8,  and  Sx-4y  =  3. 

Ans.  21a;  +  77y  -  1  =  0, 
99a; -27y- 79  =  0. 


CHAPTEK  m. 

TBANSFORMATION  OF  COORDINATES. 

48.  Certain  investigatious  are  more  easily  made  when 
the  locus  is  referred  to  a  particular  system  of  cot'jrdinates 
than  to  any  other ;  and  it  is  often  convenient  to  change  from 
one  system  to  another  by  means  of  general  formulas.  This 
may  be  done  by  finding  the  relations  between  the  courdi- 
nates  of  any  point  in  the  two  systems.  The  first  system  is 
called  the  primitive,  the  second,  the  neiv  system ;  and  the 
required  formulas  are  called  the  Equations  for  the  Transfor- 
rrwiion  of  CoifJrdinates. 

49.  Formulas  for  passing  from  one  Rectilinear 
System  to  another,  tJie  axes  being  parallel  but  the  origin  dif- 
ferent, ^y         Y     p 

Let  P  be  the  point  whose  coordinates        ^j         ^j 
in  reference  to  the  system  Y'O'X'  are 


O'A  =  X,,  OB'  =  2/„  ,  °/         °/       /^ 


and  in  reference  to  the  system  YOX,  are     o'/  c       a'  X 

Fig.  25. 

OA  =  x,AP  =  y; 

it  is  required  to  find  the  relation  between  these  coordinates. 

We  have       x,  =  OA  =0'C  +  CA  ^  O'C  +  OA, 
y,  =  OB'  =  OD  +  DB'  =  O'D  +  OB. 

Let  the  coiirdinates  of  the  origin  0  in  reference  to  0'  be 

0'G=^m,  C0  =  n, 

27 


28 


TBAN8F0BMA  TION 


[50,  51. 


and  the  preceding  equations  become 

Xi  —  m  +  Xf 

yi  =  n  +  y. 

50.  Formulas  for  passing  from  one  Rectilinear  Sys- 
tem to  another  the  origin  being  the 
same. 

Let  YOX  be  the  primitive  system,  and 
Y'OX'  the  new  system. 

Let  P  be  any  point.  Draw  PC  par- 
allel to  or,  PC  parallel  to  OY,  PB  and 
C'F  perpendicular  to  OX,  and  CD  par- 
allel to  OX.     Then 

OC  =  x,  CP  =  y,OC'  =  X,  CP  =  y'. 
Also  let  CO  =  YOX,  a  =  X'OX,  /3  =  Y'OX. 

Then  PB  =  PD  +  DB  =  PD+  C'F 

=  y'  sin  /3  +  x'  sin  a. 

But  we  also  have       PB  =  CP  sin  PCB 


=  y  sm  oj ; 
.'.  y  sin  Go  =  x'  sin  oc  +  y'  sin  /?. 


(a) 


Dropping  perpendiculars  from  P  and  C  upon  0  Y,  and  pro- 
ceeding as  before  we  find 


X  sin  00  =  x'  sin  (gj  —  a)  +  y'  sin  (&?  —  /?). 


(&) 


51.  Formulas  for  passing  from  a  Rectilinear  Sys- 
tem YOX  to  another  Rectilinear 
System  Y'OX  the  origin  being 
different. 

The  solution  consists  in  transform- 
ing the  equations  from  the  system 
FOX  to  another  parallel  system  whose 
origin  is  at  0',  and  then  to  the  system 
Y'OX'.  This  may  be  done  by  sub- 
stituting the  values  of  x  and  y  fron? 


Fig.  27. 


the  equations  of  Article  49,  which  are 


62,  53.] 


OF  COORDINATES. 


29 


n. 


in  the  equations  of  the  preceding  article.     This  being  done, 
and  the  subscripts  dropped,  we  have 


{x  —  m)  sin  oo  —  x  sin  {co  —  a)  -\-  ?/'sin  {co  —  p) 


(c) 


(y  —  n)  sin  cj  —  x'  sin  a  +  y'  sin  /J 
which  are  the  most  general  equations  for  the  transformation. 

52.  Formulas  for  passing  from,  a  Rectangular  to 
an  Oblique  System,  of  Coordinates. 

In  this  case  the  axes  of  the  primitive  system  make  a 
right  angle  with  each  other ;   hence,   oo  =         y      y 
90°.     Substituting  this  value  in  the  equa- 
tions of  Article  51,  we  have. 


X  =  m  -¥  X  cos  a  +  y  cos  /: 
y  ::^n  +  x'  sin  oc  +  y'  sin  /3 


\- 


(d)     _ 


Fig.  28. 


If  the  origin  is  the  same  in  both  systems,  we  have 
m  =  0  and  n  =  0,  and  the  Equations  for  passing  from,  Rectan- 
gular to  Oblique  CcM'dinates  the  origin  being  the  same,  become 


x  =  x'  cos  a  +  y'  cos  /?  ) 
y  =  x'  sin  o:  +  y'  sin  /?   f 


(e) 


If  the  axes  OX  and  OX'  coincide,  a  =  0  and  the  equa- 
tions become 

x  =  x'  +  y'  cos  /3  \ 
y  =  y'  sin  ^  ) 


(/) 


53.  Formulas  for  passing  from  Oblique  to  Rectan- 
g^ular  Axes,  the  origin  remaining  the  same. — Eliminating  y' 
from  equations  (e)  and  finding  x' ;  then  eliminating  x'  and 
finding  y'  gives 

X  sin  /3  —  y  cos  /? 


X  = 


sin  (/?  —  a) 


,  _  y  cos  a  —  X  sm  a 
y  ~       sin  (yS  -  a) 
which  are  the  required  equations. 


(3) 


30  .  thansformation  [54, 55. 

64.  Formulas  for  passing  from  one  Rectangular 
,     System,  to  another  the  origin  remaining 
\\    y     the   same. — For  this   case  make   m  =  0, 
n  =  0,  fc?  =  90'',  ft  =  90^  +  Lx,  in  the  equa- 
tions of  Article  51,  and  find 


x  =  x'  cos  oc  —  y'  sin  «  )         ,i\ 
y  =  x'  sin  oc  -\-  y'  cos  a  )  ' 

which  are  the  required  equations. 

If  Y'  OX'  be  the  primitive  system  and  we  pass  back  to 
the  system  YOX,  the  required  formulas  will  be  found  by 
eliminating  x'  and  y'  successively  from  equations  (h).  We 
thus  find 

x  =  X  cos  oc  +  y  sin  a  \  ,,,>. 

y'  z=  —  x  sin  «'  +  2/  cos  ac ) 

Or  these  equations  may  be  found  by  changing  a  to  —  a, 
X  to  x',  y  to  y',  x  to  x,  and  y  io  y  va.  equations  Qi).  From 
either  sets  of  equations,  we  find 


'2  . 


which  is  the  square  of  the  distance  of  any  point  from  the 
origin,  and  is  constant  in  reference  to  every  system  of  rect- 
angular axes  having  the  same  origin. 

55.  Formulas   for   passing   from   Rectangular    to 
Polar  Coordinates. 

Let  OX  be  the  initial  line,  making  an  angle  a  with  the 
Y      p  axis  of  07,   <p  the  vectorial  angle,  x—  OBy 

x'     and  y  =  PB.    We  have  from  the  figure,  the 
pole  being  at  the  origin^ 


x  =  pco^{(p  +  a))  ^  ,^ 

y  =  p  sin  (cp  +  a)  )  ' 


which  are  the  required  equations. 

If  the  initial  line  coincides  with  the  axis  of  x,  we  have 
a  =  0,  and 

x=  p  cos  (p 
y  =  p  sin  (p 


[.  0") 


55.] 


OF  COORDINATES. 


31 


If  the  pole  he  not  at  the  origin  let   its   coordinates  be  Xq, 
and  2/o>  and  we  have 


07  =  £Cj  +  p  COS  (cp  +  a)\ 
y  =  yo  +  psm  {(p  +  a)  )' 

If  the  axes  are  oblique,  making  an  an- 
gle 08?  between  them,  the  formulas  become 


{h) 


X  =  Xo-\- 


y  =  yo  + 


p  sin  [a?  —  ((^  +  a)'] 


sm  CO 


p  sin  {q)  +  a) 


sm  GO 


(0 


To  pass  from  Polar  to  Rectangular  Coordinates, 
the  pole  being  at  the  origin,  we  find  from  equations  (j) 


a?  +  y^  =  p^  (sin^  cp  +  cos'^  cp)  =  p^ ; 


cos  cp  =■ 


•'-  P=  Vx^  +  y^l 

X 


Va^  +  y 


:,  Sin  cp  = 


y 


Vix^  +  ^ 


(m) 


EXAMPLES. 

1.  To  find  the  polar  equation  to  a  right  line. 
Solution. — The  rectangular  equation  is 

y  =  mx  +  b. 

Let  the  pole  be  at  the  origin,  then  equations  ({)  give 

p  sin  (g)  +  a)  =  mp  cos  (q)  +  a)  +  i\ 

which  is  the  required  equation. 

Let  a  he.  the  angle  between  the  axis  of  x  and  the  perpendicular  (p) 
from  the  origin,  then  will 

cos  oc 

m  =  —  cot  a  =  ——. ,  and  p  =  J  sin  a. 

sin  a  '         ^ 

Substituting  the  values  of  m  and  h,  we  find 

p  cos  (p  =zp. 

If  the  vectorial  angle  be  also  measured  from  the  axis  of  .t,  and  be 


32  TRANSFORMATION  OF  COORDINATES.  [55. 

denoted  by  qi\  then  cp  =  <p'  —  a.^  and  tlie  preceding  equation  becomes, — 
dropping  the  accent, — 

p  cos  {ip  —  a)  =  p, 

which  is  the  equation  given  in  Article  3G. 

2.  To  find  the  equation  to  a  right  line  referred  to  oblique  axes. 
Solution.— The  equation  of  the  right  line  referred  to  rectangular 

axes  is 

p  =  mx  +  b. 

The  equations  for  transformation  the  origin  remaining  the  same  will 
be  equations  (e)  Art.  52 ;  hence  we  have 

x'  sin  a  +  y'  sin  yS  =  m  («'  cos  a  -\-  y'  cos  f5)  +  1  \ 
or  (sin  /S  —  m  cos  /?)  y'  =  (to  cos.  a  —  sin  a)  x'  +  &, 

which  is  the  required  equation. 

sin  9 
Let  the  axis  of  x'  coincide  with  the  axis  of  a;,  then  a  =  0,  w  =  "^^"qj 

and  we  have 

(sin  IS  cos  9  —  cos  (3  sin  9)  y'  =  sin  9. a;'  +  b  cos  9; 

or  sin  (/J  —  9)  y  =  sin  9.x'  +  b  cos  9 ; 

^  sin  9  ^  cos  9 

.     •'•  ^  ^  sin  (/^  -  9)  *'  +  sin  (/i  -  9) 

cos  9 
If  ic'  =  0,  then  y'  =—. — . ,.  _  q.  J,  which  is  the  y -inter cept ;  call  it  o, 

and  the  equation  becomes, — dropping  the  accents, — 

sin  9 
2^  =  sin(/3-9)  *  +  ^' 

which  is  the  same  as  the  equation  in  Article  27. 

3.  Given  the  point  (3,  7),  required  its  coordinates  for  another  parallel 
system  the  coordinates  of  whose  origin  are  (2,  —  2).  Ans.  (1,  9). 

4.  Given  the  point  (—  2,  3)  to  find  its  polar  coordinates,  the  pole 
being  at  the  origin  and  the  initial  line  making  an  angle  of  45°  with  the 
axis  of  X.  Ans.  p  =  V\%  cp  =  tan-'  ( —  I)  —  45°. 

5.  Given  the  point  (3,  4)  to  find  its  polar  coordinates,  the  pole  being 
at  the  point  (6,  8)  and  the  initial  line  making  an  angle  of  30°  with  the 
axis  of  X. 

6.  Transform  to  rectangular  axes  the  equation  p^  —  c^  cos  2g),  the 
origin  being  at  the  pole  and  the  initial  line  coinciding  with  the  axis  of  x. 

Ans.  {x^  +  y^y  =  c'  (x*  -  y'). 


CHAPTEE   IV. 


CONIC  SECTIONS, — THEIE  EQUATIONS  AND  PROPERTIES. 

56.  The  curves  known  as  the  Circle,  Ellipse,  Hyperbola, 
and  Parabola  are  called  Conic  Sections,  because  they  may 
be  formed  by  the  intersection  of  a  plane  with  a  cone,  as  will 
be  shown  hereafter.  But  the  investigations  in  this  chapter 
will  have  no  reference  to  the  cone.  The  equation  of  the 
curve  will  be  deduced  from  a  definition  of  some  property  ot 
the  curve.  These  curves  are  discussed  by  themselves  be- 
cause of  their  great  importance  in  the  Physical  and  As- 
tronomical Sciences. 

EQUATIONS    OF    THE    CURVES. 


TJie  Circle. 

57.  The  Circle  is  a  curve  every  point  of  which  is  at  a 
constant  distance  from  a  fixed  point  called  the  centre.  It 
may  be  described  mechanically  by  means  of  a  pencil  at  some 
point  P  of  a  string  OP  moving  around 
the  fixed  point  0.  The  locus  of  the  point 
P  will  be  a  circl3. 

To  find  the  Equation  to  the  Circle  the  ori- 
gin being  at  the  centre. 

Take  any  point  P  in  the  circumference 
and   drop  the   perpendicular  PD.      Let 
OP  =  M   the   radius   of   the   circle,  x  =  OD  and  y  =  PD, 
then  will  the  right  triangle  OPD  give 

a?  +  f  =  P^,  (o) 


34 


CONIC  SECTIONS. 


[58-60. 


which  is  the  equation  sought.     It  may  be  put  under  the 
form 


(h) 


58.  General  Equation  to  the  Circle  referred  to  rectan- 
gular coordinates. — Let  0  be  the   origin  of   coi'jrdinates,   C 

the  centre  of  the  circle,  P  any  point 
whose  coordinates  are  {x^,  y^  and  iin,  n) 
the  cot)rdinates  of  the  centre.  Accord- 
ing to  Article  49,  we  have,  for  the  trans- 
formation of  coordinates  from  G  to  0, 

x=  Xi  —  m; 
y  =  yi-n; 

which  substituted  in  equation  (a)  (Art.  57),  and  dropping 
the  subscripts,  since  x^  and  y^  are  now  general  variables, 
gives 

(x  —  rnf  +  {y  —  nf  =  B^ ;  (c) 

which  is  the  equation  sought. 

59.  Equation  of  the  Circle  referred  to  a  diameter 
and  tang:ent  at  its  vertex. — For  this  case 
m  =  H  =  OC,  and  n  =  0,  and  the  preceding 
equation  becomes 


y^  =  2Iix  —  ic^, 


(d) 


Fig.  34.  which  is  the  required  equation. 

60.  Discussion  of  Equation  (a),  Art.  57. — The  equation 
Ibo^  +  y^  =  B\     Let  x  =  0,  then 

y=±R\ 

that  is,  the  locus  cuts  the  axis  of  y  in  two  points  equidis- 
tant from  the  centre. 
Let  y  =  0,  then 

x=  ±B; 

that  is,  the  locus  cuts  the  axis  of  x  in  two  points  equidistant 
from  the  centre. 


61,  62.]  777^  CIRCLE.  35 

Solving  the  equation  for  ij  gives 


which  shows  that  for  all  values  of  ±  a:  less  than  i?,  y  has 
two  equal  and  opposite  values ;  hence  the  curve,  lies  both 
above  and  below  the  axis  of  x  and  is  symmetrical  in  respect 
to  that  axis. 

If  X  exceeds  E  the  value  of  y  will  be  imaginary,  which 
shows  that  no  part  of  the  curve  is  at  a  distance  greater  than 
R  to  the  right  or  left  of  the  centre.  Similar  properties  may 
be  shown  in  regard  to  the  axis  of  y. 

61.  To  construct  the  locus  ivhose  equation  is  oty^  +  y'^  =  25. 
Let  ic  =  0,  then  ?/ =  ±  5.  ^^y^ 

x  =  l,  ?/  — ±V24:  =  ±  4-9  nearly.       y^ 

x  =  2,  y  =  ±V^=±4:-6      "         / 

x  =  S,  y  =  ±  VT6  =  ±  4.  4 

x  =  4^  y  =  ±V9=±S.  \ 

X  =  5,  y  =  0,  \^^ 

Lay  off  above  and  below  the  centre,  ^"^  ^• 

on  the  axis  of  y,  a  distance  Ca  equal  to  5.  Then  lay  off  to 
the  right  of  the  centre  a  distance  Cb^  equal  to  1,  and  erect 
an  ordinate  biOi  above  and  below  the  axis  of  x,  equal  to  4-9 ; 
and  so  on  to  a;  =  5  =  Cb,  which  lay  off  on  the  axis  of  x.  In 
this  way  any  number  of  points  may  be  obtained  and  the 
curve  may  be  traced  through  them. 

62.  Discussion  of  Equation  (cT). — Let  x  =  0,  then  y  =  0, 
hence  the  curve  passes  through  the  origin,  (Art.  32). 

Let  y  =  0,  then  a;  =  0  and  2R,  hence  the  curve  cuts  the 
axis  of  X  in  two  points,  one  at  the  origin,  and  the  other  at  a 
distance  2Ii  from  the  origin. 

Solving  for  y^  gives 

y=±  Vi2Ii-x)x, 

hence  the  curve  is  symmetrical  in  reference  to  the  axis  of  x. 
Letting  x  be  negative,  we  have 


y=±  V{2Ii+x){-x); 


36  CONIC  SECTIONS.  [63-6G. 

wliicli  being  imaginary  shows  tliat  no  part  of  the  curve  lies 
on  the  negative  side  of  the  origin. 
Solving  for  x  gives 

which  shows  that  the  curve  is  not  symmetrical  in  reference 
to  the  axis  of  y.  If  y  be  less  than  B,  there  will  be  two  real 
values  for  x,  but  if  it  exceeds  B,  x  will  be  imaginary. 

63.  The  Centre  of  a  Curve  is  a  point  which  bisects  any 
line  drawn  through  it  terminated  by  the  curve. 

64.  A  Diameter  is  a  line  ivhich  bisects  a  system  of  parallel 
chords.  Every  diameter  passes  through  the  centre,  but  every 
line  which  passes  through  the  centre  may  not  be  a  diameter, 
as  may  be  illustrated  by  higher  plane  curves.  But  it  will 
be  found  that  every  line  which  passes  through  the  centre  of 
a  conic  section,  will  be  a  diameter.  An  Axis  to  a  Curve  is 
a  diameter  which  is  perpendicular  to  the  chords  which  it 
bisects.  The  point  where  a7iy  diameter  cuts  the  curve  is 
called  the  vertex  of  that  diameter. 

EXAMPLES. 

1.  What  are  the  coordinates  of  the  centre  of  a  circle  whose  equation 

2.  Form  the  equation  to  the  circle  whose  centre  is  (4,  —  3),  and 
radius  =  3. 

3.  Construct  a  circle  whose  equation  is  (y  —  3)*  =  4a;  —  «*. 

4.  In  the  circle  y^  +  (x  —  4)*  =  16,  what  are  the  ordinates  at  the 
point  whose  abscissa  is  7  ? 

5.  If  the  base  of  a  triangle  is  constant,  and  the  adjacent  sides  vary  in 
such  a  way  as  to  preserve  a  constant  angle  between  them ;  show  that 
the  locus  of  the  apex  is  a  circle. 

The  Ellipse. 

65.  An  Ellipse  is  a  curve,  the  sum  of  the  distances  from 
any  point  of  which  to  two  fixed  points  is 
constant. 


66.  To   trace   the    curve   mechani- 
cally, fix  two  points  of  a  string  respec- 
"FreTiT^  tively  at  the  points  F  and  F',  and  place 


67, 68.] 


THE  ELLIPSE. 


37 


a  pencil  point  at  P.  Slide  tlie  pencil  around,  keeping  the 
string  constantly  stretched ;  then  will  the  point  P  describe 
an  ellipse.     For,  in  every  position  of  P,  we  have 

F  P  +  PF—  a  constant. 

67.  Definitions. — The  two  fixed  points  F  and  F'  are 
called  the  foci  of  the  ellipse.  The 
point  C  midway  between  the  foci 
is  called  the  focal  centre,  or 
simply  the  centre.  The  lines  PF' 
and  PF,  drawn  from  any  point  P 
of  the  ellipse  to  the  foci,  are  called 
the  focal  radii.  The  diameter  A  A, 
passing  through  the  foci,  is  the  transverse  or  major  axis, 
and  the  diameter  BB'  perpendicular  to  ^^'  is  the  conju- 
gate, or  minor  axis. 

68.  Construction  of  tlie  Ellipse  by  points.— Take  any 
line  as  A' A  for  the  major  axis; 
bisect  it  at  C  and  erect  the  per- 
pendicular BB '.  If  the  length  of 
the  minor  axis  is  also  assumed, 
take  BC  equal  to  the  semi-axis 
and  with  ^  as  a  centre  and  AG  a.B 
a  radius  describe  an  arc,  cutting 
A' A  in.  the  points  F'  and  F.  These  points  will  be  the  foci, 
for  we  have 

BF'  -r  BF=  A' A  =  a  constant. 

Or  assume  the  foci  F'  and  F,  and  divide  the  distance  CF' 
into  several  parts,  Cb,  ab,  etc.,  equal  or  unequal.  Take  the 
distance  Ab  as  a,  radius  and  F'  as  a  centre,  describe  an  arc 
above  and  below  the  axis  ^4^1';  and  with  bA'  as  another 
radius  and  F  a.s  a  centre  describe  arcs  cutting  the  former 
ones  in  2  and  2.  These  will  be  points  in  the  curve.  In  this 
manner  any  number  of  points  may  be  found  through  which 
the  curve  may  be  traced.  With  the  radius  F2,  two  other 
arcs  may  be  described  having  jP  as  a  centre ;  so  that  with 
each  radius  four  arcs  may  be  described,  two  having  i^  as  a 
centre  and  two  others  having  F'  as  another  centre.    li  AC 


38  CONIC  SECTIONS.  [69. 

be  one  radius,  CA'  will  be  the  other,  and  the  intersection 
of  the  arcs  described  with  them  having  respectively  the 
centres  F  and  F'  willgive  the  extremities  B  and  B'  of  the 
minor  axis. 

An  ellipse  may  also  be  constructed  by  points  by  means 
of  its  equation.  Thus,  if  the  equation  to  the  ellipse  be  (see 
equation  {a^  of  the  next  Article), 

points  in  the  locus  may  be  found  by  assuming  values  for  y 
and  deducing  the  corresponding  values  for  x.     Thus  if 

2/=±0,±l,±2; 
then  x=  ±  2-3,  ±  2,  ±  0  ; 

which  gives  eight  points  in  the  curve.  Any  number  of  inter- 
mediate points  may  be  found  in  a  similar  manner  and  the 
corresponding  points  constructed  as  shown  for  the  circle  in 
Article  61,  through  which  the  curve  may  be  traced. 

69.  Axial  Equation  to  the  Ellipse  ;  or  in  other  words, 

_B p  to  find  the   equation  to  the  ellipse 

when  its  axes  coincide  with  the  co- 
ordinate axes. 

2c  =  FF' ;  2a  =  A' A. 

From   any  point   P  of  the    ellipse 
drop  the  perpendicular  PD ;  then  will 

x=GD,y  =  PD, 

FD  =  x-c,  F'D  =  a?  +  c 

From  the  right-angled  triangles  FDP  and  F'DP,  we 
have 


Adding,  we  have,  since  FP  +  F'P  =AA',  =  2a, 
Vix  +  cf  +  f  +  Vix  -cf  +  f  =  2a. 


70.J  THE  ELLIPSE.  39 

Freeing  this  equation  of  radicals,  gives 

(a^  -  c^)  J  +  aY  =  a^  (a'  -  (?). 

For  the  purpose  of  simplifying  this  equation,  let 

d^  —  c'  =  W, 

and  the  equation  becomes 

lr,r  +  d-ff  =  d-b',  (oi) 

which  is  the  required  equation.     Dividing  through  by  arb^ 
gives 

-^4=1,  (*.) 

or  a~V  +  b'-ff  —  1, 

which  forms  are  sometimes  more  convenient  than  equation 
(a,). 

70.  Discussion  of  Equation  (cti),  or  (&i). — Let  y  =  0, 

then 

x=  ±  a; 

hence,  the  curve  cuts  the  axis  of  x  at  two  points,  A  and  A', 
equidistant  to  the  right  and  left  of  the  centre. 
Let  X  =  0,  then 

y=±b; 

hence,  the  curve  cuts  the  axis  of  y  at  two  points  equidistant 
from  the  centre,  and  the  line  BB,  Fig.  40,  is  a  diameter. 
Solving  the  equation  for  y  gives, 


y 


±  -  Va^  —  ^ ; 


hence,  for  all  values  of  x  less  than  a,  there  are  two  equal  and 
opposite  values  for  y;  therefore  the  curve  is  symmetrical 
in  reference  to  the  transverse  axis.  If  xya,  y  is  imagi- 
nary. In  a  similar  way  we  may  show  that  the  curve  is  sym- 
metrical in  reference  to  the  minor  axis. 


40 


CONIC  SECTIONS.  [71,  72. 

The  value  of  ¥  given  above  is 


tliat  is,  tlie  distance  from  either  ex- 
tremity of  the  semi-conjugate  axis 
to  either  focus  equals  the  semi- 
major  axis ;  a  result  which  was  deduced  in  Article  68  directly 
from  the  definition  of  the  curve. 

71.  A  General  Equation  to  the  Ellipse,  tl^e,  coordinate 
axes  being  parallel  to  the  axes  of  the 
ellipse. — Let  0  be  the  origin  of  co- 
ordinates, C  the  centre  of  the 
ellipse  whose  coordinates  are 

OF=m,UC=n; 

P  anj  point  whose   coordinates 

OI)  =  x„DP  =  y^; 

then,  according  to  Article  49,  we  have  for  transferring  the 
origin  from  C  to  0 

X  =  Xi  —  m; 

y  =  yi-n; 

which  substituted  in  equation  (oi)  (Art.  69)  give,  after  drop- 
ping the  subscripts, 

¥  (x  -  my  +  a'(y-  nf  =  a%' ;  (c,) 

which  is  the  required  equation. 

72.  Rectangular  Equation 
to  the  Ellipse,  the  origin  being 
at  the  vertex  of  the  major 
axis,  the  axis  of  x  coinciding  ivith 
the  major  axis. — In  this  case  we 
have 


Fig.  42. 


m  =  A'C=  a,  n  =  0; 


73,  74.] 


THE  HYPERBOLA. 


41 


hence,  equation  (cj)  becomes, 


y^  =^(2aic  —  x^), 


(d.) 


which  is  the  required  equation. 


EXAMPLES. 

1.  Find  the  axial  equation  of  the  ellipse,  the  major  axis  being  7  and 
the  minor  axis  3. 

2.  Construct  the  equation  of  the  ellipse,  the  foci  being  (3,  0),  (—  3,  0) 
and  the  vertex  of  the  semi-conjugate  axis  being  (0,  4). 

3.  The  vertex  of  the  semi-transverse  axis  being  (5,  0)  and  one  of  the 
foci  (3,  0),  form  the  equation  to  the  ellipse. 

4.  In  an  ellipse  the  sum  of  the  focal  radii  is  18,  and  the  square  of  the 
distance  between  the  foci  less  8^  is  36,  find  the  equation  to  tlie  ellipse. 

5.  Find  the  axial  equation  of  an  ellipse  which  shall  pass  through  the 
points  (-  3,  2),  (-4,  1). 

6.  If  the  origin  is  at  the  vertex  of  the  conjugate  axis,  and  the  axis 
of  y  coincides  with  that  axis,  show  that  the  equation  to  the  ellipse  be- 

comes  x^  =~{2hy  —  y^). 

7.  The  lower  end  of  a  bar  whose  length  is  21 
slides  upon  a  horizontal  plane  while  its  upper  end 
slides  along  a  vertical  plane ;  what  curve  will  a  fly 
describe  that  remains  upon  the  bar  at  a  distance  d 
from  the  centre  while  the  bar  slides  down  ? 

8.  Tlie  ordinate  of  a  circle  «*  +  y^  =  r^  is  in- 
creased by  a  line  equal  in  length  to  n  times  the 
ordinate;  show  that  the  locus  of  the  end  of  the 
ordinate  thus  increased  is  an  ellipse. 


The  Hyperlola. 

73.  An  Hyperbola  is  a  curve  the  difference  of  the  dis- 
tances of  any  point  of  which  from  two  fixed  'points  is  constant. 

74.  Mechanical  Construction.— Let  F  and  F'  be  the 
fixed  points.  Let  the  ruler  F'B  be 
pivoted  at  F'.  Take  a  string  whose 
length  BPF  is  less  than  F  B,  and 
fasten  one  end  at  F  and  the  other  at 
some  point  B  on  the  ruler.  With  a 
pencil  point  at  P  keep  the  string 
pressed  against  the  ruler  as  the  latter  fio.  44. 


42  CONIC  SECTIONS.  [75-78. 

is  turned  about  the  point  F' ;  then  will  the  point  P  describe 
the  arc  of  an  hyperbola.     For  \l  PC  =  PF  we  have 

BF'  -{BP  +  FC)  =  CF'\ 
or  {BF'  -  BP)  -PC=  OF'. 

But  BF'  -BP  =  PF'; 

.:PF'-PC=CF\ 
or  PF'  -  PF  =CF'=a  constant ; 

hence  the  locus  of  P  will  be  an  hyperbola.  By  pivoting  the 
ruler  at  F,  another  branch  of  the  curve  may  be  described. 

75.  Construetion  by  Points. — Join  i^'i^  and  let  A' A 

be  the  constant  difference  be- 
tween PF'  and  PF,  P  being 
any  point  on  the  curve.  Make 
F'V=A'A.  Withi^'asacen- 

*~a    tre  and  a  radius  F'a  greater 
than  F  A '  describe  arcs  above 
and  below  the  line  A  A .  Do  the 
Fig.  45.  Same  witli  i^  as  a  Centre.   Then 

with  a  radius  equal  to  Va,  and  with  ^  as  a  centre  describe 
arcs  intersecting  the  former  ones.  Do  the  same  with  i^ '  as  a 
centre.  The  intersection  of  the  corresponding  arcs  will  be 
points  in  the  curve,  for,  by  the  construction,  PF '  —  PF  = 
F'V—B.  constant.  In  this  way  any  number  of  points  may 
be  located  through  which  the  curve  may  be  traced. 

76.  Definitions. — The  fixed  points  F '  and  F  are  called 
the  foci  of  the  hyperbola.  The  line  A  A  is  the  transverse 
axis.  The  lines  F  P  and  FP  are  the  focal  radii.  The 
points  A'  and  A  are  the  vertices  of  the  curve.  The  point  C, 
midway  between  the  foci,  is  the  centre,  or  focal  centre,  of 
the  curve. 

77.  A  Branch  of  a  Curve  is  a  continuous  portion  of 
the  curve.  The  hyperbola  has  two  branches,  the  ellipse  and 
circle  each,  one  branch.     Some  curves  have  many  branches. 

78.  Equation  of  the  Hyperbola  referred  to  its 
axes. — The  origin  will  be  at  the  centre  C,  midway  between 


79] 


THE  HYPERBOLA. 


43 


the  foci  F  and  F '.  Let  the  axis  of 
X  coincide  with  the  axis  CA  of  the 
curve,  and  from  any  point  P  of  the 
curve  drop  the  perpendicular  PD ; 
then  will 

x  =  CD;  y  =  DP. 

Let  ^^'  =  2a;  i^i^'  =  2c, 

then     CA  =  a\  CF=c;  F'D  =  x  +  c\  FD  =  x  -  c. 

The  right  angled  triangles  F  DP  and  FDP,  give 


Fig.  46. 


F'P  =  \/{x  +  c)-  -i-  ?/-'  ; 


FP  =.-  V(x  -  c)-  +  f  ; 
and  from  the  definition  of  the  curve  we  have 


FP  -FP  =  2a=  V{x  +  cf  +y^-  V{x  -ef  +  f. 

Clearing  of  radicals  and  reducing  gives 

(c^  —  a-)  a^  —  d^if  =  a^  {<?  —  dF) ; 

and  making  (?  —  c^  =  6^, 

the  equation  becomes 

V'j?  -  ay  =  (rh\  (oa) 

which  is  the  required  equation.     It  may  be  written,  by  di- 
viding through  by  a'6^ 

or,  a'^J^  —  1-^1^  =  1. 

Equation  (62)  is  the  equation  of  the  curve  in  terms  of  its 
intercepts. 

79.  Discussion  of  Equation  (a.).  — Let  y  =  0,  then 
x=  ±a;  hence  the  curve  cuts  the  axis  of  .r  in  two  points 
A  and  ^'  equidistant  from  the  origin.  A  A'  is  a  diameter, 
and  also  an  axis  of  the  curve,  and  C  is  the  centre. 


44 


CONIC  SECTIONS. 


[79. 


Let  3^  =  0,  then  ^Z  =  ±h\/  —1,  whicli,  being  imaginary, 
shows  that  the  curve  does  not  cut  the  axis  of  y.  The  position 
of  this  imaginary  line  corresponds  with  that  of  the  conjugate 
axis  of  the  ellipse ;  hence,  by  way  of  analogy,  and  in  an 
analytical  sense,  we  speak  of  the  conjugate  cuis  of  the  hyper- 
bola and  represent  its  half  length  by  &,  the  real  part  of  the 
preceding  expression. 

From  the  expression  <?  —  c?  —  }?,  given  in  the  preceding 
Article,  we  have 

^  ■=  O?  ■\- }? ; 

hence :  Tlie  distance  from  the  centre  to  either  focus  equals  the 
hypothenuse  of  the  triangle  constructed  on  the  semi-axes. 
If  a  =  &  in  Eq.  {a^  we  have 


x'  —  y*  —  cf 

which  is  called  the  equation  to  the  equilateral  hyperbola.     It 
resembles  somewhat  the  equation  to  the  circle. 

If  an  hyperbola  be  constructed  having 
its  transverse  axis  coincident  and  equal  to 
BB',  the  axis  conjugate  to  AA,  it  is  called 
a  conjugate  hyperbola  in  reference  to  the 
former  one.  Either  hyperbola  is  called  a 
conjugate  in  reference  to  the  other.  The 
equation  to  the  conjugate  hyperbola  will 


Fig.  47. 


be 


y 


a» 


s-  -— =1 

■JO  o  —   -«-• 


Solving  equation  (a^)  in  reference  to  y  gives 
y  =  ±---Vo^-a\ 


which  shows  that  for  all  values  of  x  less  than  a,  y  is  imagi- 
nary, and  for  all  values  greater  than  a,  y  has  two  real  values, 
equal  and  opposite.  Hence  the  curve  lies  both  above  and 
below  the  axis  of  x,  and  is  symmetrical  in  reference  to  that 
axis. 


80,81  ]  THE  HTPERBOLA. 

Solving  in  reference  to  .>',  we  have 


45 


=  ±j-^/fTh\ 


which  is  real  for  all  values  of  y ;  hence,  for  every  positive  or 
negative  value  of  y,  there  are  two  real,  equal  and  opposite 
values  of  x,  or:  The  curve  is  symmetrual  in  reference  to  the 
ascisofy. 

80.  A  General  Equation  to  the  Hyperbola,  the  axes 
of  coordinates  being  parallel  to  the 
axes  of  the  curve. 

Let  the  coordinates  of  the 
centre  in  reference  to  the  new 
axes  be 

OE=m;  EC=n; 

and  of  any  point  P,  (x^,  y^.  Then, 
according  to  Article  49,  we  have 


x=  Xi  —  m, 
y  =  yi~n; 


which  values  substituted  in  equation  (a^)  give, — after  drop- 
ping the  subscripts, — 


I^{x-  mf  -a'iy-  nf  =  a'V", 


(c,) 


which  is  the  required  equation. 

81.  Equation  to  the  Hyperbola,  the  origin  being 
at  the  left  vertex  of  the  Curve,  the  axis  of  x  coinciding 
with  the  transverse  axis. — We  will  have  m  =  a  and  w  =  0  in 
equation  (^),  and  the  equation  becomes 


f=l{^-.%^). 


(dz) 


46 


come  SECTIONS. 


[83,  83. 


EXAMPLES. 

1.  Find  the  axial  equation  to  the  hyperbola,  the  transverse  axis  being 
7  and  the  distance  between  the  foci  being  9. 

2.  The  square  of  the  conjugate  axis  being.  —  9,  and  the  transverse 
axis  being  4,  find  the  axial  equation  to  the  liypeibola. 

3.  Find  the  equation  to  the  hyperbola,  the  origin  being  at  the  left 
vertex,  the  major  axis  being  12,  and  the  distance  between  the  foci  16. 

4.  In  a  given  hyperbola,  the  difference  of  the  focal  radii  =  8,  and 
the  difference  between  the  squares  of  that  difference  and  the  distance 
between  the  foci  =  —  9 ;  find  the  equation. 

5.  The  ordinate  of  an  hyperbola  is  prolonged  so  as  to  equal  the  cor- 
responding focal  radius ;  find  the  locus  of  the  extremity  of  the  prolonga- 
tion. 

6.  Show  that  the  equation  of  the  hyperbola  whose  real  axis  is  the 
conjugate  axis  of  the  hyperbola  h^x^  —  a^y^  =  a'b^,  is  a^y^  —  &*a;*=  a* J" 


The  Parabola. 

82.  The  Parabola  is  a  curve  every  point  of  which  is 
equally  distant  from  a  fxed  liTie  and  fixed  point. 
If  F'  is  the  fixed  point  and  GE  the  fixed  line, 
then  for  any  point  P  on  the  curve  we  have 
PF=PG. 

83.  To  describe  the  curve   mechanically. 

Let  GE  be  a  fixed  line  and  F  a  fixed  point. 

Take  a  board  or  ruler  having  a  right  angle  at 

C,  and  attach  one  end  of  a  string  at  B. 

Stretch  the  string  along  the  edge  BC, 

and  swing  the  end  C  around  to  the 

point  F  and  secure  it  there.     With  a 

pencil  point  P  press  the  string  against 

the  edge  BC  while  the  ruler  is  moved 

along  the  fixed  line,  the  pencil  being 

permitted  to  move  along  the  edge  BC; 

the  curve  traced  by  the  point  will  be 

.     For  the  length  of  the  string  will  be 

BP  +  PC=BP+PF, 

and,  subtracting  BP  from  both  sides  of  the  equation,  we  hare 

PC=  PF, 
which  is  the  condition  required  by  Article  82. 


Fig.  50. 

the  required  curve 


84-86.] 


THE  PARABOLA. 


47 


Fig.  51. 


-Assume  a 


Fio.  52. 


84.  Definitions.  -The  fixed  point  i^is  called  the  focus. 
The  fixed  line  6^0  is  called  the  directrix. 
The  straight  line  through  the  focus,  per- 
pendicular to  the  directrix,  is  called  the 
axis  of  the  pnrahola  ;  the  point  A  where 
the  axis  cuts  the  curve  is  called  the  vertex* 
and  the  line  FP,  drawn  from  the  focus  to 
any  point  of  the  curve,  is  called  the  focal 
radius. 

85.  To  Construct  a  Parabola  by  Points, 
fixed  line  G  0  and  a  point  F.  Through  F 
draw  a  line  FO  perpendicular  to  G  0.  Bi- 
sect OF  Sit  A,  then  will  A  be  one  point  in 
the  curve,  for  it  is  equidistant  from  the 
fixed  line  and  point.  To  find  another  point, 
assume  any  radius  as  FP  greater  than  FA, 
and  with  i^  as  a  centre  describe  an  arc  and 
intersect  it  by  a  line  drawn  parallel  to  OG 
and  at  a  distance  from  it  equal  to  FP ;  then  will  the  point 
of  intersection  P  be  a  point  on  the  curve.  For  by  the  con- 
struction we  will  have  GP  =  J^F.  In  a  similar  manner  any 
number  of  points  may  be  found. 

The  points  may  also  be  found  by  means  of  the  equation 
of  the  curve.  See  equation  (cZg)  of  the  following  Article,  and 
Article  61. 

86.  Equation  to  the  Parabola  the  Origin  being  at 
the  Vertex  of  the  Curve. — Let  the  axis  of  x  coincide  with 
the  axis  of  the  curve,  and  the  axis  of  y  be  tangent  to  it. 

We  have,  (Fig.  51), 

X  =  AD,  y  =  PD, 
and  letting  p  be  the  constant  distance  OF,  we  have 

0A  =  AF=lp; 
GP=OD  =  x  +  ^p  =  FF; 
FD  =  x-lp', 
and  the  triangle  FDP  gives 

FP^  =  FD^  +  DP"', 


48  CONIC  SECTIONS.  [87, 88. 

or  {x  +  Ipf  =  {x-  ip)2  +  f ; 

wliicli  reduced  gives 

2/2  =  2jox  (di) 

for  the  required  equation. 

87.  Discussion  of  Equation  (ds). — If  x  =  0,  y  =  0, 
hence  the  curve  passes  through  the  origin  of  coordinates. 
This  also  follows  from  the  fact  that  the  equation  has  no 
absolute  term,  (Art.  32). 

Solving  for  y  gives 

y  =  ±  V2px 

which  is  real  for  every  positive  value  of  x,  and  gives  two 
equal  and  opposite  values  for  y.  Therefore,  the  curve  is 
symmetrical  in  reference  to  the  axis  of  x.  If  x  be  negative, 
y  will  be  imaginary,  hence  the  curve  extends  only  in  the 
positive  direction  of  x.  It  has  only  one  branch.  As  x 
increases  indefinitely,  y  also  increases  indefinitely,  hence  the 
branch  is  infinite  and  the  curve  is  not  reentrant.  Strictly 
speaking,  therefore,  it  cannot  have  a  centre,  but,  for  the  sake 
of  symmetry  in  th.e/o7'7n  of  expression,  we  say  that  its  centre  is 
at  an  infinite  distance  from  the  vertex.  Since  all  diameters  of  a 
curve  pass  through  the  centre,  it  follows  that  oR  diameters  of 
a  parabola  are  parallel  to  the  axis. 

Let  x  =  Ijp,  the  abscissa  of  the  focus,  then 

y=p, 

hence  the  ordinate  at  the  focus  equals  twice  the  distance  of 
the  focus  from  the  vertex ;  and  : —  The  double  ordinate  at  the 
focus  equals  four  times  the  distance  of  the  focus  from  the  vei'tex. 

88.  Equation  to  the  Parabola  the  Origin  being  at 
any  Point  and  the  axis  of  x  paraUd  to  the  axis  of  the  curve. — 
Let  the  vertex  of  the  curve  in  reference  to  the  new  origin  be 
(m,  n),  X  and  y  the  coordinates  of  any  point  when  the  origin 
is  at  ^ ;  cci,  y^  the  corresponding  coordinates  of  the  same 
point  when  the  origin  is  at  0,  then  according  to  Article  49, 
we  have 

x  =  i\-m,  y  r=zy^-n. 


89, 90.]  TEE  CONIC  SECTIONS  COMPARED.  49 

which  in  equation  (c/3)  give,  after  dropping  the  subscripts, 

{y  -  nf  -2p{x-m)=  0.     (03) 

If  the  origin  be  at  B,  the  inter-    y 
section  of  the  axis  and  directrix, 
we  have  n  ■=  0,  and  m  =  BA  = 
^p,  and  equation  (rg)  becomes 


B 

A 

r      ^ 

p 

V 

p 

y^  =  2p(x-  Ip), 


Fig.  53. 


which  is  t1i£.  equation  to  the  parabola  referred  to  its  axis  and 
directrix.  If  the  origin  be  removed  to  the  focus,  the  axes 
remaining  parallel,  we  have 

y^  =  2p  (j:  +  Ip). 

EXAMPLES. 

1.  Find  the  rectangular  equation  to  a  parabola  in  which  the  coor- 
dinates of  the  focus  are  (4,  0),  the  origin  being  at  the  vertex. 

2.  The  origin  being  at  the  vertex,  find  the  rectangular  equation  to  a 
parabola  which  shall  pass  through  the  point  x  =  4,  y'  =  16. 

3.  The  distance  of  the  focus  from  the  directrix  being  8,  find  the 
equation  of  the  parabola  referred  to  the  axis  and  directrix. 

4.  Show  that  the  locus  of  the  centre  of  the  circle  which  is  tangent  to 
the  axis  of  y  and  passes  through  a  fixed  point  on  the  axis  of  x  is  a  para- 
bola. 

89.  The  equations  to  the  circle,  ellipse,  and  hyperbola 
have  been  given  when  the  origin  is  at  the  centre  of  the  curve, 
but  no  such  equation  can  be  given  for  the  parabola,  since  it 
has  no  centre  at  a  finite  distance,  as  shown  in  Article  87. 


The  Equations  of  the  Conic  Sections  Compared. 

90.  When  the  Curves  are  referred  to  their  Axes 
we  have  found  their  equations  to  be  ;  see  equations  (6),  (61), 
(62),  of  the  preceding  pages ;  for 


The  Circle 


f_ 


(&) 


50 

come  SECTIONS 

The  EUipse 

The  Hyperbola 

The  Parabola 

(No  finite  equal 

[91 

(6i) 


A  cofaparison  of  these  equations  shows  that,  if  in  the 
equation  of  the  ellipse  a  =  h  =  R,\i  becomes  the  equation  of 
the  circle.  Also,  in  the  equation  of  the  ellipse,  if  we  make 
W  =  —Ir'  it  becomes  the  equation  of  the  hyperbola. 

We  shall  therefore,  when  the  curves  are  referred  to  their 
axes,  determine  a  required  property  for  the  ellipse,  and 
deduce  the  corresponding  property  for  the  circle  by  making 
a  =  6  in  the  result,  and  for  the  hyperbola  by  making  h'  = 
-h\ 

[We  may  readily  conceive,  how,  for  a  given  major  axis  these  curves  pass 
from  one  to  the  other.  Thus,  in  a  circle  the  foci  may  be  considered  as  con- 
secutive with  the  centre ;  and  if  the  foci  separate  from  each  other,  the 
major  axis  remaining  constantly  equal  to  the  diameter  of  the  circle,  we  have 
an  ellipse  whose  minor  axis  &  is  constantly  diminishing  ;  and  when  the 
foci  reach  the  ends  of  the  major  axis  the  ellipse  becomes  a  right  line,  and 
when  they  pass  those  points,  the  ellipse  changes  to  an  hyperbola  in  which 
the  conjugate  axis  will  increase  as  the  foci  are  more  and  more  separated.] 

91.  "WTien  the  Curves  are  referred  to  their  axes  and 
a  tangent  at  the  left  vertex,  we  have  ;  see  equations  {d), 
(di),  (d^),  (ds) ;  for 

The  Circle  y^  =  2Bx  -o?\  (d) 

TheElUpse  y^  =  '^x-^i^;  {d,) 

The  Hyperbola     y"^  = x  +  -jCC^;  {d^ 

The  Parabola        y^  =  Ipx ;  (c^) 
all  of  which  are  included  under  the  general  form 

2/2  =  Pa;  +  Itx\  {D) 


93.]                     THE  come  SECTIONS  COMPARED.  61 

In  this  equation  R  is  the  ratio  of  the  squares  of  the  semi- 
axes,  and  P  is  called  the  principal  parameter  of  the  curve 
(Arts.  95,  and  145).     Hence  for  the  Conies 


the  value  of 

and  the  ratio  of  the 

the  parameter. 

squares  of  the  semi-axes,  1 

for  the  circle 

P  =  2i?,     .     .     .     .     i?  =  — 1; 

ellipse 

P-2^'                         R-      ^^• 

hyperbola 

a'                           or' 

parabola 

P=2p,    .     .     .     .     R  =  0. 

A  comparison  of  these  equations  also  shows,  that  if,  in 
the  equation  of  the  ellipse,  we  make  a=  b  =R  it  becomes 
the  equation  to  the  circle ;  and  if  6^=  —  h'  it  becomes  the 
equation  to  the  hyperbola.  Hence,  when  the  curves  are 
referred  to  corresponding  vertices  and  axes,  the  properties 
of  the  circle  and  hyperbola  may  be  deduced  from  those  of 
the  ellipse,  by  substituting  for  the  value  of  b  in  the  results  for 
the  ellipse,  the  values  given  above  for  the  respective  curves. 

92.  General  Equation  of  the  Second  Degree. — All 
the  equations  of  preceding  Articles  are  of  the  second  degree, 
and  it  will  be  shown,  (Art.  177),  that  every  equation  of  the 
second  degree  represents  a  conic. 

The  general  equation  of  the  second  degree  may  be  written 

Ax^  +  2Exy  +  By'  +  ^Gx  +  2Fy  +  C=0. 

It  is  written  this  way  so  as  to  conform  to  certain  modern 
usage.  It  is  shown  in  Article  178  that  this  equation  will 
represent 

a  circle  if  A  =  B  and  H=0; 

an  ellipse  if        H^  —  AB  is  negative ; 

an  hyperbola  if  H^  —  AB  is  positive ; 

a  parabola  if      E^  —  AB  =  0. 


52 


CONIC  SECTIONS. 


EXAMPLES. 
Determine  to  what  locus  the  following  equations  belong: 

a;2  +  j,2  +  Sy  —  4a;  +  2  =  0. 
3a!*  +  nxy  +  7y«-  2a;  +  4y  -  8  =  0. 
2x^  +4j^  +  yi  +  3a;  -  2y  -  1  =  0. 

5a;*  -  Zxy  -  2y*  -  2j!  -  7  =  0. 

4a?*  +  12a;y  +  9y*  —  a;  +  lOy  —  4  =  0. 

2y*  _  3j;  +  4  =  0. 

Eccentricity. 

93.  The  Eccentricity  of  a  Conic  Section  is  the  ratio 

of  the  distance  of  the  focus  from  the  centre  to  the  length  of  the 
semi-transverse  axis.     In  other  words,  if  the  semi-transverse 


Fig.  55. 

axis  were  unity,  the  eccentricity  would  be  the  distance  of 
the  focus  from  the  centre. 
Let  e  =  the  eccentricity ; 

c  =  the  distance  of  either  focus  from  the  centre,  = 
CF=CF'; 


then  we  have  for  the  dUpse, 


_CF  _c  _Vor^         /       62 
^~CA~a~       a       ~V  a'' 


and  changing  6^  to  —  6^  we  have  for  the  hyperbola, 


(1) 


(2) 


93.]  ECCENTEICITT.  53 

and  making  a  =  co,  (Art.  87),  we  have  for  theparahola 

e  =  1 ;  (3) 

and  making  a  =  b,  -we  have  for  the  circle 

e  =  0. 

From  these  we  observe  that  for 

The  Circle  e  =  0; 

The  Ellipse  e  <  1 ; 

The  Hyperbola  e  >  1 ; 

The  Parabola  e  =  1. 

From  equation  (1)  we  find 

I'  =  a^  (1  -  e^,  (4) 

from  which  the  value  of  the  semi-minor  axis  may  be  found 
in  terms  of  the  eccentricity  and  the  semi-major  axis. 

If  e  =  0,  &  =  a. 
Ife<l,  b<a. 

Ife>l,  b=V^^a(e'-l)K 

Ife  =  l,  6  =  0  X  00,  which  is  indeterminate,  although  in 
this  case,  we  know  from  Article  87,  that  6  =  go. 

Remark.  — Here  we  have  another  analogy  between  the  several  conic  sec- 
tions. To  illustrate,  Let  i^ and  F'  be  the  foci  of  an  ellipse,  Cits  centre,  and 
AY  &  tangent  at  the  principal  vertex. 
Suppose  that  the  foci  approach  uniformly 
towards  6,  the  curve  being  of  such  varia- 
ble dimensions  as  to  remain  constantly 
tangent  to  AY;  all  such  curves  will  be 
ellipses,  but  when  F  and  F'  become  con- 
seputive  to  C,  the  curve  becomes  a  circle 
also  tangent  U)  AY,  and  having  zero  for 
its  eccentricity,  and  ^C  for  its  radius.  Be- 
ginning again  with  the  foci,  as  shown  in 
the  figure,  let  the  right  focus  F'  move  to  the  right,  the  focus  ^remaining 
fixed,  and,  therefore,  the  distance  AF  remaining  constant ;  then,  for  all 
finite  values  of  FF '  the  curve  will  be  an  ellipse,  the  eccentricity  of  which 
may  be  determined  as  shown  above.  But  as  the  centre  C  moves  to  the  right, 
the  distances  AG  and  FG  approach  equality,  and  for  FG  =  <x>,  we  have 
AO  =  00 ;  hence,  ultimately, 

FG_. 
AG-  ' 


Fio.  56. 


54 


CONIC  SECTIONS. 


[9a 


and  the  finite  portion  of  the  curve  which  is  that  in  the  vicinity  of  A,  be- 
comes a  parabola. 

When  the  eccentricity  exceeds  unity,  the  vertex  of  the  curve  which  dis- 
appeared at  the  right  of  A,  will  reappear  at 
A' ,  at  the  left  of  A,  the  distance  A  A'  at  first 
being  —  oc,*  but  constantly  decreasing  (nu- 
merically) as  the  eccentricity  increases.  The 
finite  portion  of  this  curve  is  an  hyperbola. 
This  statement  is  here  made  without  proof, 
but  it  can  be  shown  to  be  true  in  a  beautiful 
Fio.  57.  manner  by  means  of  a  cone  and  cutting  plane, 

(see  Art.  185).  If  A'  continues  to  approach  A,  and  finally  coincides  with  it, 
the  eccentricity  becomes  infinite,  and  the  hyperbola  becomes  the  straight  line 
AP  ;  for  the  difference  of  the  distances  PF 
and  PF  will  be  constant,  being  zero.  Hence  a 
straight  line  is  one  limit  of  the  hyperbola.  If 
now  the  foci  F  and  F  approach  J.  aud  finally 
coincide  with  it,  the  two  branches  of  the  hyper- 
bola become  any  two  right  lines  passing  through 
it ;  which  is  another  limiting  case  of  the  hyper- 


bola. 


93a.  Eccentric  Angle. 

D'       p. 


If  a  circle  be  described  on  the  major  axis 
of  an  ellipse,  and  an  ordinate  BP'  be 
erected,  and  CP,  CP'  be  drawn,  the 
angle  P  'CA  is  called  the  eccentric  angle 
in  reference  to  the  point  P. 

hetx'  =  CB,  y'  =  PB,   cp=P'CA, 
CA  =a,  CD=  6, 


then    x'  =  a  cos  cp, 


y'  =h  sin  q). 


the  value  of  x'  being  deduced  directly 
from  the  figure,  and  of  y'  by  substitut- 
ing the  value  of  a; '  in  the  axial  equation 
Fig.  59.  to  the  ellipse  and  solving  fory'. 

The  auxiliary  angle  (p  is  useful  in  solving  certain  problems  (see  exam- 
ple 64,  p.  141). 


*  [There  are  many  cases  in  which  a  function  passes  from  +  oc  to  —  co 
for  a  finite  increase  of  the  variable.  Thus,  in  the  equation  y  =  tan  x,  y 
passes  from  0  to  +  oo,  as  a;  increases  from  0  to  ^ff  ;  and  as  x  passes  \it,y 
changes  from  -i-  oo  to  —  oo.     Similar  conditions  exist  for  the  equations 

y=seca!;y  =  cota!;  y  =  -; —  ;  etc.] 


94,  95.] 


LATUS  RECTUM. 


65 


Latus  Rectum. 

94.  The  Latus  Rectum  of  a  Conic  Section  is  the 
dovUe  ordinate  to  the  transverse  axis  through  the  focus.  This 
is  also  called  the  principal  parameter,  or  parameter  of  tlie 
curve. 

95.  Value  of  the  Latus  Rectum. — For  the  ellipse,  make 
x=  ±c  in  equation  {a^,  (Art.  69),  and  solve  for  y ;  double 
the  ordinate  thus  found  will  be  the  y 

value  sought.    We  have 


f=^A^'-c) 


=— r,  (since  a-  —  c^  =  V) : 
a 


Fig.  60. 


.-.  2/=±- 


(1) 


in  which  the  positive  value  is  the  ordinate  above  the  major 
axis,  and  the  negative  value  the  part  below.  As  these  are 
equal  in  value,  the  double  ordinate  at  the  focus  will  be, 
numerically,  twice  the  value  of  either ; 

.:PP'  =  ^=-^^=2a{l-^).  (2) 

a  *Za 

Similarly,  for  the  hyperhohy 
1? 


y='f- 


.\PP 


,_{2hf_{BB'f 
2a  ~  A  A 


=2a{^-l);  (3) 


Fio.  61. 


that  is :  In  the  ellipse  and  hyper- 
bola, the  latus  rectum  is  a  third  proportional  to  the  transverse 
axis  and  its  conjugate. 

These  expressions  are  also  true  when  a  =  b  =  B,  hence, 
true  for  the  circle ;  and,  by  analogy,  we  may  say  that  the 
principal  parameter  of  a  circle  is  any  diameter. 


66  CONIC  SECTIONS.  [06. 

Far  the  Parabola,  make  x=^pin.  equation  {d^,  Article 
Y  86,  which  equation  is 

P/  y^  =  2px, 

and  we  find 
y-—^  2y  =  PP'  =  2p  =  4:OF; 

P^      that  is :  The  latus  rectum  of  a  paralola  eqvxds  four 
^^'  '^^'       times  the  distance  of  the  focus  from  the  vertex  of  the 
curve. 

96.  Remark. — ^We  now  see  that  the  coefficients  of  x  in 
the  equations  of  Article  91  are  the  parameters  of  the  respec- 
tive curves,  and  are  represented  by  P  in  equation  {D).  We 
see  from  inspection  that  we  may  write,  for  the  square  of  the 
ratio  of  the  semi-axes, 

in  which  the  parameter  P  may  be  constant,  while  R  varies 
inversely  as  the  semi-major  axis  a.  Tlie  equation  to  the 
conic  may,  therefore,  be  written,  the  origin  being  at  the  prin- 
cipal vertex, 

f=Px^-?.rj^=  ~{2ax + x% 
2a         2a 

P 

If  a  =  00 ,  —  =  0,  and  we  have  y^  =  Px,  which  is  the  equa- 
tion of  the  parabola. 


EXAMPLES. 

1.  What  is  the  principal  parameter  to  the  curve 

3««  +  4y«  -  12  ? 

2.  What  is  the  principal  parameter  to  the  curve 

2a:«  -  7y»  =  8  ? 
8.  What  is  the  principal  parameter  to  the  cur^e 
4y«  =  12a!  ? 


97.] 


OF  0RDINATE8. 


67 


B    P 


Of  Ordinates. 

97.  Let  the  point  P  be  {x',  y'),  and  P',  {x",  y"),  then 
equation  (hi),  (Art.  69),  gives  for  the 
ellipse : 

hence,  y'^  :  y'"-  : :  {o?  -  x"-)  :  {o?  -  x"^) 

:  {a  +  «')  (a  —  x)  :  (a  +  a?")  {a  —x") 
:  AD. DA  :  AD. DA-, 

which  is  also  true  for  the  circle  and  hyperbola,  since  it 
does  not  contain  h.  Hence,  for  the  circle,  ellipse,  and  hyper- 
bola :  Tlve  squares  of  the  ordinates  to  the  major  axis  are  propor- 
tional to  the  products  of  the  corresponding  segments  into  which 
the  axis  is  divided  by  the  ordinates. 

From  the  preceding  proportion,  we  find 

AD.  DA     AD.  DA 


DF" 


DP" 


which  is  a  convenient  form  for  memorizing. 

If  a  circle  be  described  on  the  major  axis  of  an  ellipse, 
and  an  ordinate  P  'D  be  erected, 
we  have,  from  the  equations  of 
the  curves, 


P'D''=a^-a?,PD''=-^{a^-a?)', 

.\P  D  '.  PD  ::  a  :  b, 

or  2a  :  26; 

that  IB :  If  a  circle  be  described  on 
the  major  axis  of  an  eHipse^  and 


Fio.  64. 


58  CONIC  SECTIONS.  [98, 

a  common  ordinate  he  erected  to  that  axis,  the  ordinate  of  the 
circle  vnll  le  to  the  ordinate  of  the  ellipse,  as  tJie  major  axis  of  tJw 
ellipse  is  to  its  minor  axis. 

A  similar  proportion  may  be  found  if  a  circle  be  described 
on  tlie  minor  axis,  and  an  ordinate  be  erected  to  tliat  axis. 

For  the  Parabola  we  have 

y'^  =  2px';  y"^  =  2px"; 

.•.  y  ^  :  y  ^  ::  X    :  x 

tliat  is  :  The  squares  of  the  ordinates  to  the  axis  of  the  parahohf 
are  as  the  corresponding  abscissas. 

Of  Intersections  and  Tangents. 

98.  To  find  the  points  of  Intersection  of  a  Right 
Line  with  an  Ellipse. — The  axial  equation  to  the  ellipse 
is 

h^x-  +  a V  =  ^^^^  i 

and  the  equation  of  the  right  line 

y  =  mx  +  d ; 

{d  being  used  instead  of  the  h  heretofore  given  so  as  not  to 
confound  it  with  the  h  in  the  equation  of  the  ellipse).  The 
coordinates  of  the  points  of  intersection  must  satisfy  both 
equations ;  hence,  considering  the  equations  as  simultaneous, 
and  eliminating  y,  we  find 

a?^^2~J  J  amd  +  ^d'm'd?  -  {¥  +a'm')  (d'  -  b')    |.   (1) 

If  the  quantity  under  the  radical  be  positive,  there  will 
be  two  real  points  of  intersection;  if  it  be  negative,  the 
points  will  be  imaginary,  or,  in  other  words,  the  line  will  not 
intersect  the  curve ;  but  if  the  radical  part  is  zero,  there  will 
be  only  one  point,  and  the  line  will  be  tangent  to  the  curve. 
The  last  condition  requires  that  we  have 

a'm'd' =  (Jr- +  a'm')  {d?-W)', 

.'.  d  =  VoWTPT  (2) 


93.]  OF  INTERSECTIONS  AND   TANGENTS.  59 

wMcli  substituted  in  the  preceding  equation  of  the  right 
line,  gives 

y  =  mx  +  ^/d^w?  +  [?■  ;  (3) 

and  every  equation  of  this  form  is  the  equation  of  a  tangent 
to  an  ellipse. 

For  the  Hyperbola,  this  becomes  by  changing  W'  to 
-^.  

y  =  mx  +  's/c^rri^  —  }/  ;  (4) 

and  for  the  Circle,  making  a^  =  1/^  —  H'  in  equation  (3), 


y  =  mx  +  R  Vw-  +  1 .  (5) 

For  the  Parabola  we  may  find 

These  are  called  the  3Iagical  Equations  to  the  tangent. 

The   magical  equation  of   the   tangent  to  the  circle  is 
easily  found  geometrically. 

Let  BT  he  the  tangent  at  S,  CS  the 
radius,  and 

m  =  tan  BTC  =  tan  BCS  =BS---CS; 
then  "^ 


sec  SCB  =  \/tang-  +  l  =  Vm'  +  1 ; 

.'.BC=BV^mFri;  ''"•''• 

which  is  the  intercept  on  the  axis  of  y ;  hence  the  equation 
of  TB,  which  is  of  the  form 

y  =  mx  +  h, 


becomes  y  =  mx  +  B  Vm^  +  1 

as  given  above. 

Similarly,  the  points  of  intersection  of  any  two  curves 
may  be  found  by  considering  their  equations  as  simulta- 


60  C02s''IG  SECTIONS.  [99. 

neons  and  eliminating  first  one  variable  and  tlien  the  other. 
Curves,  or  lines,  represented  by  equations  of  the  second 
degree  are  called  curves  of  the  second  order.  Two  curves  of 
the  second  order  will,  in  general,  intersect  each  other  in 
four  points ;  for  the  elimination  between  two  general  equa- 
tions of  the  second  degree  gives  rise  to  an  equation  of  the 
fourth  degree  of  which  there  will  be  four  roots. 


EXAMPLES. 

1.  Find  the  points  of  intersection  of  the  lines 

x^  +  y^  =  R"";  y  =  Zx-4:. 

2.  Find  the  points  of  intersection  of  the  lines 

y*  =  4a; ;  y  =  —  2a;  +  5. 

3.  Find  the  points  of  intersection  of  the  curves 

y*  =  9a;;  x^  +  y^  +  Zx—  y  =  16. 

4.  Find  the  points  of  intersection  of  the  parabola  y^  —'Zx  +  10,  and 
the  hyperbola  10.r*  —  4i/*  =  8. 

5.  Find  the  equation  of  a  tangent  to  the  ellipse  Zz*  +  ly*  =  8,  the 
tangent  being  inclined  45°  to  the  axis  of  x. 

Of  Tangents  and  Suitangents. 

99.  A  Tangent  to  a  curve  is  a  right  line  which  passes 
through  two  consecutive  points  of  the  curve,  or  which  touches 
the  curve  in  one  point.     LetP  and  P'  be  any  two  points  of 

a  curve  through  which  a  secant 
is  passed.  Let  the  secant  turn 
about  the  point  P,  the  point  P ' 
remaining  in  the  secant  and  mov- 
ing towards  P ;  when  P '  becomes 
consecutive  to  P,  or  when  it  falls 
upon  and  coincides  with  P,  the 

^«-  ««•  line  r  Pr  will  be  tangent  to  the 

curve.* 


*  In  elementary  geometry  a  tangent  is  defined  to  be  a  line  Avhich  touches 
a  curve  in  one  point  only.     While  this  definition  answers  for  many  curves. 


100.]  OF  TANGENTS.  61 

100.  Equation  of  a  Tangent  to  the  Ellipse.— Take 
the  axial  equation 

Jrx^  +  a^y^  =  a^h^ ; 

and  let  a  secant  be  passed  through  the  points  {x',y'),  (»",  y"). 
The  coordinates  of  these  points  must  satisfy  the  equation  of 
the  ellipse ;  hence  we  have  the  equations  of  condition 

b^x"^  +  a"y"^  =  a-lr ; 
h^x"^  +  ay^  ^  a-h\ 

Subtracting  one  from  the  other  and  factoring,  we  have 

y"  -  y  ^_^l_.  ^  +  ^'  .  (1) 

x'  —X  .  a-    y'  +  y" 

The  equation  to  a  right  line  passing  through  these  points 
is,  (Art.  40), 

,       y"  —  y',  ,x 

y-y  =  x" -X  (^~^)> 

in  which  substitute  the  value  of  the  left  member  of  equation 
(1),  and  we  have 

^  a^   y  +  y 

When  the  points  through  which  the  secant  passes  become 
consecutive,  we  have 

x'  =  x",  and  y'  =  y"  ; 

including  the  conic  sections,  it  is  not  general ;  for  tliere  are  many  curves  in 
which  a  line  tangent  at  one  point  of  a  curve  may  cut  it  in  several  other  points. 
The  definition  in  the  text  involving  two  consecutive  points,  is  not  only 
the  more  general,  but  it  is  the  most  useful  in  making  investigations. 
Students  at  first,  are  generally  slow  to  admit  that  a  line  passing  through 
two  points,  though  they  be  consecutive,  is  a  tangent,  for  they  affirm  that 
sucli  a  line  is  a  secant.  But  a  tangent  may  be  considered  as  a  special  case 
of  a  secant,  as  a  circle  is  a  special  case  of  an  ellipse  ;  and  when  the  secant 
passes  through  two  consecutive  points  its  position  will  differ  from  that  tan- 
gent which  passes  through  only  one  of  them  by  less  than  any  assignable 
quantity.  The  expressions  therefore  for  the  slope  of  such  a  secant  will  be 
the  same  as  for  the  tangent,  (see  Art.  33).  We  may  say  then  that  a  tangent 
is  a  secant  which  passes  through  two  consecutive  points  of  a  curve. 


62  CONIG  SECTIONS.  [101. 

and  these  values  substituted  in  tlie  preceding  equation  give 

h^x'x  +  aYy  =  ^y'x^  +  <^V  =  «'^' ;  (3) 

for  the   required  equation.     Dividing  through  by  a^6%  it 
becomes 


(e) 


the /orwi  of  which  is  similar  to  that  of  the  ellipse. 


101.  Equation  of  the  Tangent  to   the  Circle,    the 

origin  being  at  the  centre. 


Make  a  =  1  =  Biii  equation  (e),  and  we  have 


XX     yy_ 


+ 


=  1, 


(ed 


or 


x'x  +  y'y  =  B^ ; 
for  the  required  equation. 

[This  result  may  easily  be  deduced  directly  from  the  figure.     For  we 
7'  have 

m  =  ianPTX=  -  GT-i-  CT=  —  GD-^PD=z 


and 

CT'  :  CP  '.:  CP  '.  PD;  .:  CT'  =  ^' 

y 

The  equation  of  the  line  TT '  will  be  of  the 


y         y' 


Fig.  67. 


form 


which  becomes 
as  given  above.] 


xaf  -hyy'  =  R*, 


103-104  "• 


OF  TANGENTS. 


63 


102.  Equation  of  the  Tangent  to  the  Hyperbola, 

the  curve  being  referred  to 
its  axes.  —  Writing  —  W 
for  y^,  in  equation  (e)  in 
Article  100,  gives 


^2 


yy 


1,      (e.) 


Fig.  68. 


whicli    is    the    required 
equation. 

103.  Equation  of  the  Tangent  to  the  Parabola, 

the  curve  being  referred  to  its  axis,  and  the  y 

tangent  at  its  vertex. — The   equation   to 
the  curve  will  be 


y"  =  2px ; 

and  the  eqvMtions  of  condition  for  a  secant 
will  be 

y'^  =  Ijpx  ;  y"'^  —  2px" ; 


FiQ.  69. 


from  which  we  find 


y"-y'  ^    2i7 


x'  —  x       y"  +  y'. 

But  the  equation  of  the  right  line  passing  through  two 
points  is,  (Art.  40), 

^ — ^=-^77 — ^=-^,-^- — ^, ,  (from    the   pre- 
X  —  x      x  —  x      y   -^  y 

ceding  equation).     Making  y"  =  y',  we  find 

y'y=p{x  +  x'),  (ea) 

which  is  the  required  equation. 

These  equations  may  be  discussed  in  the  same  manner 
as  the  equation  to  any  other  right  line. 

104.  Intercepts  of  the  Tangent  to  a  Conic  Section. — 
For  the  intercept  on  the  axis  of  x,  make  ?/  =  0  in  equations  (e), 


64  CONIC  SECTIONS.  [104; 

(ei),  (62),  (63)  and  we  have  for 

theeUipse   .    .  CT ~', 

IP 

the  circle  .   .    .   CT  =  —7-; 

X 

the  hyperbola .  CT=—t  ; 

the  parabola .  .  CT  =  —  x' . 

Tojlnd  the  intercept  on  the  axis  of  y  make  ic  =  0  in  the  sam© 
equations,  and  we  find  for 

the  ellipse.    .  CT' =K; 

^  y 

the  circle   .    .  CT '  = — r> 

y 

the  hyperbola  CT'=-^\ 

x' 
the  parabola .  CT'=p—f. 

if 


EXAMPLES. 

1.  Find  the  equation  of  the  tangent  to  the  ellipse  3a;*  +  5^*  =  10 
at  a  point  whose  abscissa  is  1 ;  also  find  its  intercepts  and  construct  the 
line. 

2.  Find  the  equation  of  the  tangent  to  the  circle  x^  +  y^  —  13  at  the 
point  (3,  —  2),  and  construct  the  line. 

3.  Find  the  equation  of  a  tangent  to  the  hyperbola  3a;*  —  4?/*  =  12 
at  the  point  where  the  latus  rectum  cuts  the  curve. 

4.  Find  the  equation  of  a  tangent  to  the  parabola  ?/*  =  4a;,  and  deter- 
mine its  equation  when  it  is  inclined  30°  to  the  axis  of  x. 

^.Observe  that,  in  equation  {e^\-^=^tan.of  the  inclination,  or  the  slope 

as  it  is  sometimes  called.) 

104a.  To  find  the  eqitation  of  the  tangent  to  an  ellipse  in  terms  of  the 
eccentric  angle. 

The  coordinates  of  the  point  will  be  (Art.  93«), 

a^  =  a  cos  q),  y'  =b  sin  9 ; 


105.]  OF  8UBTANGENT8.  65 

and  these  values  in  the  equation  of  the  tangent  give 
asm  q).  y  +  b  cos  q).x  =  ab, 
for  the  required  equation. 
The  intercepts  will  be 

a  J 


y 


cos  q>  sm  q} 

105.  The  Subtangent  is  the  distarwe  between  the  foot 
of  the  ordinate  of  contact  and  the  foot  of  the  tangent.  The 
foot  of  the  tangent  is  the  point  where 
it  intersects  either  axis  of  coordi- 
nates, but  unless  otherwise  men- 
tioned, the  foot  will  be  considered 
as  on  the  axis  of  x.  Let  FT  he  a. 
tangent  at  the  point  P,  then  will  PI)  fvg.  7o. 

be  the  ordinate  of  contact,  D  the  foot  of  the  ordinate,  T  the 
foot  of  the  tangent,  and  DT  the  subtangent.  The  subtan- 
gent  is  the  projection  of  the  tangent  on  the  axis  of  x.  Let 
x^  —  CT,  the  intercept  of  the  tangent  on  the  axis  of  x,  and 
x'  =  CDf  the  abscissa  of  contact ;  then  will 

xV  ^    ^-^  CC^  ~~*  Ou  • 

Substituting  the  value  of  x^  =  CT,  from  the  preceding 
Article,  we  have  for  the  ellipse,  hyperbola,  and  cird£. 


DT=^-x'  = 


,  _d  -  x'^  _(ct+  x'){a  -a^) 
x'      ~  x' 

AD.  DA 


CD 


(1) 


hence :  The  subtangent  of  the  ellipse,  hyperbola,  and  circle 
is  a  fourth  proportional  to  the  segments  of  the  major  axis  formed 
by  the  ordinate  of  contact,  and  the  abscissa  to  the  point  of  con- 
tact. 

In  the  hyperbola  the  axis  is  not  divided  by  the  ordi- 
nate, but  in  order  to  generalize  the  principle  we  consider 
that  the  axis  is  prolonged,  and  define  a  segment  as  the  dis- 
5 


66 


come  SECTIONS. 


[106,  107. 


tance  from  the  foot  of  the  ordinate  to  either  extremity  of  the 
axis. 

For  the  parabola  we  have  CD  =  x,  (Fig.  69),  and  CT  = 
—  x\  (Art.  104) ;  hence  disregarding  the  sign  of  CTy  we  have 

TD  =  x'  +  x'  =  1x' ; 

hence  :  Tim  suhtangent  of  the  parabola  is  bisected  at  the  vertex  of 

the  curve. 

[Obs. — Considering  CI)  as  positive,  GT  will  be  negative, 

and  we  have . 

I)T=  +  CD-CT 

=  x'  —  {—x')  =  2x', 
as  before.] 

Length  of  the  Tangent. 
106.  In  all  the  Conies  we  have,  (Figs.  68,  69,  70), 


Also, 


TP  =  VTD''  +  DP^=  VTB""  -\-y'K 

TF=-X^. 
sm  T 


EXAMPLES. 

1.  What  is  the  length  of  the  subtangent  to  the  ellipse  ^x^  +  ly^  =  35 
at  a  point  whose  abscissa  is  2  ?  and  what  is  the  length  of  the  tangent  for 
the  same  point  ? 

2.  What  is  the  length  of  the  subtangent  to  the  circle  a;^  +  y*  =  25  at 
the  point  ( —  3,  —  4)  ?  and  what  is  the  length  of  the  tangent  ? 

3.  What  is  the  length  of  the  tangent,  and  of  the  subtangent  to  the 
parabola  y^  =  9a;  at  the  point  (4,  6)  ? 

107.  In  a  Conic  Section  the  Acute  Angles  between 
the  tangent  and  focal  radii,  at  any  point,  are  equal  to 

each  other. — Let  P  be  any 
point  on  the  ellipse,  PjTthe 
tangent,  PF'  and  PF  the 
focal  radii ;  then  will 

TPF=  T'PF'. 


Fl».  71. 


The  equation  to  the  right 
line    passing    througrh  the 


107.]  OF  THE  TANGENT.  67 

points  F  and  P  will  be,  (Art  40), 

y-y  =^^^('^-^)y 

in  which  the  point  F  is  (y"  =  0,  x"  =  c),  and  P,  {x',  y') ;  and 
hence  the  equation  becomes 

y-y'  =  ^^'i^-^')- 

Similarly,  the  equation  of  the  line  F'P  will  be 
V  —  V  =  — ^—7  (x  —  x). 

The  equation  of  the  tangent  line  FT  is,  (Eq.  (e)  Art.  100), 

Wx         }? 
a^y         y  ' 

hence,  according  to  Article  44,  we  have 

¥x'         y' 

i^FFT= ±-4i£-=-|g±^,  =  -^,.  (1) 

1  + 


y'     f     ^\  cy'ia^  +  ex)         cy' 

c+x'\     a^y'J 


and 

_ya;'  y' 

^FPT=         ^'y      "^^"'      =^f--^l  =  ^„       (2) 

1      y'   ( J^^'\   ^(«-^)  ^ 

—  c-^x'\     a^y'l 

But  from  the  figure  we  have 

tan  T'PF'  =  tan  (180°  -F'PT)  =  -  tan  i^'TT' ; 
which  compared  with  equations  (1)  and  (2)  gives 

T'PF  =  FPT.  (3) 


68 


CONIC  SECTIONS. 


[107. 


For  the  Hyperbola  we  also 
have 

T'PF'  =  FPT. 

In  this  curve  the  focal  radii  lie  on 
opposite  sides  of  the  tangent,  while 
for  the  ellipse  they  are  on  the  same 
side. 

In  the  Parabola,  the  equation  of  the  tangent  TP  is, 
(Art.  103,  Eq.  {&,), 


Fig.  72. 


in  which 


y  =  ^>{^+^')y 


^,  =  tan  PTF. 

y 


The  equation  of  the  line  Pi^is,  (Eq.  (5),  Art.  40), 


y-y 


y  -y 


J  {x  —  x') ; 


in  which  y"  =  0,  x"  =  hp=  CF,  x  =  CD,  y'  =DP  ;  and  the 
equation  becomes 

hence,  according  to  Article  44,  we  have 


y 


tan  TPF= 


P. 

hp-^'    y' 
y  ^hp 


p  . 


(4) 


that  is 


TPF=PTF=EPF': 


(5) 


PF'  being  a  diameter  passing  through  P.  In  order  that 
the  loording  of  the  proposition  at  the  beginning  of  this  Arti- 
cle shall  apply  strictly  to  the  parabola,  it  is  necessary  to 


108.]  OF  THE  TANGENT.  69 

consider  the  diameter  PF'  as  a  focal  radius,  as  we  have  pre- 
viously done,  (Art.  87,  and  Remark  in  Art.  93). 
Since  the  angle  T  equals  TPF,  we  have 

TF^FP,  (6) 

and  the  triangle  TFP  is  isosceles. 

For  the  Circle,  the  focal  radii  coincide  and  form  the 
radius,  hence  the  tangent  will  be  perpendicular  to  the  radius, 
as  is  well  kno^vTi. 

In  the  ellipse  and  parabola  the  tangent  bisects  the  exter- 
nal angle  formed  by  the  focal  radii,  but  in  the  hyperbola  the 
internal  angle  is  bisected. 

108.  Constniction  of  the  Tangent  to  a  Conic  Sec- 
tion.— A.  Let  the  tangent  be  drawn  through  a  point  on 
THE  Curve.  I''.  By  means  of  focal  radii.  On  the  focal 
radii,  or  on  one  of  them  and  on  the  other  prolonged  if  neces- 
sary, take  equal  distances 

PH=  PF, 


Pig.  74.  Fig.  75.  Fio.  76. 


and  join  F  and  H.  Through  P  draw  a  line  P  T  perpendicu- 
lar to  FH,  and  it  will  be  the  tangent  required ;  for  it  bisects 
the  angle  FPH  formed  by  the  focal  radii. 

[The  mode  of  constructing  the  curves  by  means  of  a  string  leads  to  the 
same  construction.  For  the  string  from  one  focal  radius  will  be  elongated 
an  amount  equal  to  that  by  which  the  other  is  shortened  ;  hence  PH  will 
represent  the  rate  of  shortening  of  PF',  and  Pi?* the  corresponding  rate  of 
elongating  PF,  and  the  direction  of  the  pencU  point  will  be  that  of  the 
resultant  of  these  two  rates,  and  will,  therefore,  bisect  the  angle  between 
them.] 


70 


CONIC  SECTIONS. 


[loa 


Y     , 

^^^ 

^ 

A  T 


Fig.  77. 


By  means  of  the  Subtangent.     According  to  Article 
105,  the  subtangent   is   indepen- 
dent of  the  minor  axis.    To  apply 
the  method  to  the  ellipse,  draw 
a  circle  on  A'Aas  &  diameter,  and 
-X    through  the  given  point  F  erect 
an  ordinate  P  PD,  and  at  P '  in 
the  circle  draw  a  tangent  P '  T. 
The  line  TP,  passing  through 
the  points  T  and  P,  will  be  the  tangent  required. 

For  the  parabola,  let  P  be  the  point ;  drop  the  perpen- 
dicular PD  on  the  axis  CD,  and  take 
CT  equal  to  CD  on  the  axis  prolonged ; 
then  will  P  T,  drawn  through  P  and  T, 
be  the  tangent  required.  For  the  sub- 
tangent TD  will  be  bisected  at  the  ver- 
tex, (Art.  105). 

S''.  By  means  of  the  intercept  of  the  tan- 
gent on  the  axis  of  x,  especially  for  the 
This  method  is  equivalent  to  the  former  one, 
for  the  subtangent  is  deduced 
at  once  by  means  of  the  inter- 
cept. Let  P  be  the  point  of 
which  CD  is  the  abscissa  =  a?'. 
With  C  as  a  centre,  and  CD 
as  a  radius,  describe  an  arc, 
and  at  the  vertex  A  erect  an 
ordinate  AM,  prolonging  it 
till  it  intersects  the  arc  DM 
at  M.  Join  C  and  31,  and  with  a  radius  CA,  and  centre  (7, 
describe  an  arc  intersecting  CM  at  N.  Drop  the  perpendic- 
ular NT;  then  will  the  line  PT,  drawn  through  P  and  T,  be 
the  tangent  required.     For  we  have 


Fia.  78. 


hyperbola. 


Fig.  79. 


or 


CM  :  CA  ::  {CN=  CA)  :  CT; 
x'  :  a  ::  a  :  CT 


.-.  CT  =  -,; 

X 


108.] 


OF  THE  TANGENT. 


71 


which,  according  to  Article  104,  is  the  intercept  of  the  tan- 
gent on  the  axis  of  x. 

In  a  manner  quite  similar,  the  inter ce]3t  on  the  axis  of  y 
may  be  found. 

According  to  the  last  equation,  the  value  of  CT  dimin- 
ishes as  x'  increases,  and  if  x  —  go,  CT  —  0 ;  hence,  as  x  in- 
creases indefinitely,  the  intercept  approaches  the  centre  C  as 
a  limit,  and  the  point  T,  for  the  right-hand  branch  of  the 
curve,  can  never  be  at  the  left  of  C. 

4^.  By  means  of  a  Normal. — See  Article  118. 

5".  By  means  of  conjugate  diameters. — See  Article  126. 

B.    To    DRAW    A    TANGENT    TO    A    CONIC    SECTION    THROUGH   A 

POINT  WITHOUT  THE  CURVE. — For  the  ellipse,  with  one  focus 
i^'  as  a  centre,  and  a  radius 
equal  to  the  major  axis,  de- 
scribe an  arc ;  and  with  the 
given  point  P  as  a  centre,  and 
radius  PF  equal  to  the  dis- 
tance of  P  from  the  other 
focus,  describe  another  arc 
intersecting  the  former  in  the 
points  M  and  N.  Join  F'M 
and  F'N  and  the  points  P ' 
and  P  ",  where  these  lines  in- 
tersect the  curve,  will  be  the 
tangent  points,  and  the  lines  PP '  and  PP"  will  be  the  tan- 
gents required.     For,  by  the  construction, 

F'P'  +  P'M=A'A', 

and  because  the  point  /"  is  on  the  curve 

F'P'  +  P'F=AA; 

.'.  P'F=P'M; 

also  PF=PM; 

therefore  two  points,  P  and  P',  are  equally  distant  from  M 
and  F,  hence  the  line  PP '  will  be  perpendicular  to  F3I,  and 
bisect  the  angle  FP'M,  and  therefore  is  a  tangent  re- 
quired.    Similarly,  the  line  PP  "  is  also  a  tangent. 


Fig.  80. 


72 


CONIC  SECTIONS. 


[109 


Fio.  81. 


P^-i 


For  the  hyperbola,  let  P  be  tlie  point.     Use  the    same 

language  and  correspond- 
ing quantities  as  just  given 
for  the  ellipse.  The  proof 
is  also  the  same,  only  ob- 
serving that  P  M  becomes 
negative  for  the  hyper- 
bola. 

For  the  parabola  with  the 
given  point  P  as  a  centre, 
and  radius  PF,  {F  being  the  focus),  describe  an  arc  cut- 
ting the  directrix  MN  in  the  points  31 
and  N.  Draw  MP'  and  NP"  perpen- 
dicular to  the  directrix,  and  the  points 
P'  and  P"  where  they  intersect  the 
curve  will  be  tangent  points,  and  PP ', 
PP"  will  be  the  required  tangents. 
For  by  the  construction  P  is  equidis- 
tant from  M  and  F,  and  because  P '  is 
on  the  curve,  it  is  equally  distant  from 
the  same  points,  hence  PP'  will  be 
perpendicular  to  the  line  joining  M  and  F,  and,  therefore, 
will  bisect  MP  F,  which  is  the  required  condition.  Simi- 
larly, PP"  is  a  tangent. 

(To  make  this  construction  appear  the  same  as  for  the 
ellipse  and  hyperbola,  it  is  only  necessary  to  consider  one 
focus  as  infinitely  distant.  The  arc  described  from  that 
focus  as  a  centre,  will  be  the  straight  line  MN  at  the  left  of 
vertex,  a  distance  equal  to  that  of  the  vertex  from  the  focus. 
And  the  focal  lines  drawn  through  M  and  N  to  the  remote 
focus  will  be  parallel  to  the  axis  of  the  curve.  In  the  pre- 
ceding cases,  no  tangent  will  be  possible,  if  the  point  is 
within  the  curve.) 

C.   To  DRAW  A   TANGENT  TO  A   COmC  SECTION  PAEALLEL  TO  A 

GIVEN  UNE. — See  Article  126. 

Normals  and  Subnormals. 
109.  The  Normal  to  a  curve  is  the  perpendiciHar  to  the 
tangent  at  the  point  of  tangency  and  limited  by  one  of  the  axes. 


Fig.  82. 


110.  J 


NORMALS. 


73 


It  will  be  understood,  unless  otherwise  stated,  that  the  foot 
of  the  normal  is  the  point  where  the  normal  intersects  the 
axis  of  X.  Thus,  PN  is  the  nor- 
mal at  the  point  P. 

The  Subnormal  is  the  pro- 
jection of  the  normal  on  the 
axis  of  X.  Thus,  ND  is  the 
subnormal. 

I'lG.  S3. 

110.  Equation  of  the  Normal  to  the  Ellipse.— Let 
x'y'  be  the  point  P  through  which  the  normal  is  drawn. 
The  equation  to  a  line  passing  through  this  point  will  be 


y  —  y'  =m{x  —  X 


(1) 


The  equation  to  the  tangent  through  the  same  point,  Art. 
100,  Eq.  (e),  may  be  written 


y  = 


dy  y 


(2) 


b^x' 


in  which ,^  is  the  tangent  of  the  angle  which  the  tan- 

a-y  ° 

gent  line  makes  with  the  axis  of  x,  (Art.  28),  and  the  condi- 
tion which  will  make  the  former  line  perpendicular  to  the 
latter  is  (Art.  45), 

(3) 


m 


1       1  _ay. 


and  this  value  in  equation  (1)  gives 

y-y  ^ftv^^"'^^' 


(4) 


which  is  the  required  equation.     Clearing  of  fractions  and 
dividing  by  x'y'  gives 


ax 
x' 


y' 


a^-l?=(?; 


(5) 


which  is  also  the  equation  of  the  normal,  and  is  similar  in 
form  to  the  equation  of  the  tangent. 


74 


CONIC  SECTIONS. 


[111-114 


111.  The    equation   to  the  Normal  of  the   Circle 

becomes,  by  making  a  =  'b=Rin.  equation  (5) ; 


X       y 


(6) 


and  since  it  lias  no  absolute  term  it  expresses  the  well-known 
fact  that  the  normal  to  the  circle  (which  is  the  radius)  passes 
through  the  centre  of  the  circle. 

112.  The  equation  to  the  Normal  of  the  Hyper- 
bola becomes,  from  Eq.   (5),  (writing 

^^  +  ^  =  a'-^1^  =  (?.         (7) 
X        y 

113.  The  equation  to  the  Nor- 
mal of  the  Parabola  is  found  to 
be,  by  a  process  similar  to  that  in 
Article  110, 


y  I         '\ 
y-y=-^{x-x). 


(8) 


Fio.  85. 


113a. — To  find  the  equation  to  the  normal  of  an  ellipse  in  terms  of  tht 
eccentric  angle. 

The  equation  to  the  tangent  may  be  written,  (Art.  104  a). 

b  cos  cp  b 

y= ; ■X+-. ; 

a  sin  (p        sin  <p 

hence  the  equation  to  the  normal  passing  through  the  jwint  (a  cos  q>,  b  sin  (p) 

will  be 

,    .  a  sin  <p  , 

—  0  sin  9)  =  ^^-^^^ — ■  (x  —  a  cos  ^) ; 


6  cos  g} 


which  reduces  to 


a  sec  <p.x  —b  cosec  <p . y  =  a*  —  6'  =  c*. 

114.  Intercepts  of  the  Normal. — To  find  the  inter- 
cept on  the  axis  of  x,  make  ?/  =  0  in  equations  (5),  (6),  (7), 
(8),  and  the  corresponding  value  of  x  will  be  the  intercept 
required.    In  this  way  we  find,  for 


114a.]  SUBNORMALS.  75 

the  ellipse  .  .  .  CN=  —5  x'  —^x'  ; 

the  circle.  .  .  .  CN=0', 
the  hyperbola  .  CN  =  e^x  ; 
the  parabola .  .  AN  —x'+p; 

in  which  e  is  the  eccentricity,  (Art.  93),  and  j^  is  one-half  the 
parameter  of  the  axis.  The  value  of  CN  for  the  ellipse  and 
hyperbola  appear  to  be  the  same,  but  they  are  not  really  so, 
since  e  for  the  ellipse  is  less  than  unity,  and  for  the  hyper- 
bola, greater  than  unity. 

The  intercept  on  the  axis  of  y  may  be  found  by  making 
x  —  0  and  solving  for  y  in  the  same  equations. 

1 14a.  The  Length,  of  the  Subnormal  is  the  difference 
between  CD  and  CN.  Using  only  the  positive  results,  since 
it  is  the  numerical  value  which  is  sought,  we  have 

DN  =  x'^  CN; 

in  which  substitute  the  value  of  CN  from  the  preceding  Ar- 
ticle, and  we  have  for 

the  ellipse  .  .  .  DN=  x'—  e^x'=  (1  —  e-)x'=-^x' ; 
the  circle.  .  .  .  DN=  x' ; 
the  hyperbola.  DN=—^x'; 

the  parabola.  .  DN=p. 

In  the  ellipse,  circle,  and  hyperbola,  the  length  of  the 
subnormal  varies  directly  as  the  abscissa,  while  in  the 
parabola  it  is  constant.     In  the  ellipse  if  x  -- 0,  DN=0; 

andifa?'=<7,  DN=-;   hence  the  subnormal  is  greatest  at 

€t 

the  extremity  of  the  major  axis,  and  at  that  point  equals  the 
ordinate  through  the  focus,  (Art.  95).  For  the  hyperbola  the 
subnormal  is  a  minimum  at  the  vertex  of  the  curve,  at  which 

point  x'=  a,  and  we  have  DN=  -=  half  the  latus  rectum, 

(Art  95).    From  this  point  it  increases  indefinitely  with  x'. 


76  come  SECTIONS.  [115-117. 

115.  Length  of  the  Normal. — In  all  the  curves,  we 
have 

Normal  —  V{Subnormaiy  +  (Ordinate)^ 


=  y/PD-  +  DN\ 

116.  Distance  of  the  foot  of  the  Normal  from  either 
Focus. — 1°.  The  distance  from  the  focus 

^^         •  wMcli  is  on  tJie  positive  side  of  tJie  origin. 
The  required  distance  is  NF,  F  be- 
ing at  the  focus,  and  we  have 
y^     c  T   V  F  D    N" 

FN=CF^CN', 

Fig.  86. 

in  which  substitute  the  value  of  CF  (equal  ac  for  all  but  the 
parabola,  (Art.  93),  and  equal  Ip  for  the  parabola,  (xA.rt.  85),) 
and  ON,  (Art.  114:),  and  we  have  the  following  numerical 
values  for  the  distance  of  the  foot  of  the  normal  from  the  focus 
on  the  positive  side  of  the  origin,  for 

the  ellipse.  .  .FN=e{a  —  ex'),(e<l); 
the  circle   .  .  .  FN=  0,  (e  =  0) ; 
the  hyperbola  FN=  e  {ex'  —  a),  (e>l) ; 
the  parabola . .  FN  =  a;'  +  \p,  (e  =  1). 

Similarly,  for  the  distance  on  the  negative  side,  since  cc' 
becomes  negative,  we  have  numerically,  for 

the  ellipse .  .  .  F'N —eia  -{-  ex')\ 

the  circle   .  .  .F'N=Q', 

the  hyperbola  F'N=e  {ex  +  a) ; 

the  parabola..  i^'iV^=  oo  on  the  positive  side. 

[As  there  is  no  real  second  focus  to  the  parabola,  this 
expression  is  only  one  oi  form.  It  is  necessarily  positive 
since  no  part  of  the  curve  lies  on  the  negative  side  of  the 
origin.] 

117.  In  any  Conic  Section  the  Normal  bisects  the 


118] 


NORMALS  AND  SUBNORMALS. 


11 


Angle  between  the  focal  radii,  or  between  one  focal 
radius  and  the  other  prolonged.— The  angle  between  the 
normal  and  the  tangent  is  right,  and  it  has  been  proved  in 
Article  107  that  the  acute  angles  betAveen  the  focal  radii 
and  the  tangent  are  equal  to  each  other ;  hence  taking  each 
of  the  latter  from  a  right  angle  leaves  the  remaining  angle 
on  one  side  equal  to  the  remaining  angle  on  the  other  side 
of  the  normal. 

It  will  be  observed  that  in  the  ellipse  and  parabola,  the 
normal  bisects  the  internal  angle  formed  by  the  focal  radii, 
and  in  the  hyperbola,  the  external  angle. 

118.  Construction  of  the  Normal. — A.  Let  the  point 

THROUGH  WHICH  THE  NOEMAL  IS  TO  BE  DEAWN  BE  ON  THE  CUEVE. 

1°.  Construct  a  tangent  to  the  curve  at  that  point,  (Art.  108), 
and  at  the  point  erect  a  perpendicular  to  the  tangent;  it 
■will  be  the  required  line. 

2°.  Bisect  the  angle  between  the  focal  radii,  and  it  will  be 
either  a  normal  or  a  tangent,  (Arts.  108  and  117).  If  the 
latter,  erect  a  perpendicular  as  stated  in  the  preceding  case. 

S°.  By  means  of  the  subnormal.  For  the  ellipse  and  hy- 
perbola we  proceed  as  follows : 


Fig.  87. 

We  hays,  (Art.  114a), 


a^ 


in  which  x  =  CD.     To  construct  this  take  the  lines  CA  and 
CBy  Fig.  89,  making  any  convenient  angle  with     b 
each  other.     Make  CA  —  a,  CB  =  b,  and  CD  -= 
x.    Join  B  and  A  and  draw  i?e/parallel  to  AB. 
Take  CG  equal  to  CJ  and  draw  GK  parallel 


78  CONIC  SECTIONS.  [118. 

to  AB ;  CK  will  be  the  required  subnormal.     For  we  have 

CK     _  CJ 
^^^  CJ=CG~CD* 

•  nir—  ^^_  ^v 

which  is  the  value  sought.  The  distance  CK  being  laid  off 
from  D,  Figs.  87,  88,  in  the  proper  direction  gives  the  point 
JV,  and  PjY  will  be  the  required  normal. 

For  the  parabola  let  P  be  the  point.  Drop  the  perpen- 
dicular I*B  to  the  axis  of  the  curve, 
and  lay  off  BN  equal  to  one-half  of 
the  parameter  of  the  curve  ;  then  will 
the  line  PN  be  the  required  normal. 

[This    method   also  furnishes  an 

easy  mode  of  drawing  a  tangent  to  a 

*'io-90-  parabola  at  a  given  point.     For  the 

tangent  will  be  perpendicular  to  the  normal  at  the  point  P.] 

4:th.  By  means  of  the  intercept  on  the  axis  of  x, — For  the 

ellipse  and  hyperbola,  the  value  of  the  intercept  is,  (Art 

114), 

a' 

which  expression  is  of  the  same  form  as  that  for  BN  in  the 
preceding  case  ;  hence  it  may  be  constructed  in  the  same 
B^  manner.     Making  ^(7=c,  we  have  CK—  the 

value  of  the  intercept,  which   equals    CN  in 
Figs.  87,  88,  92. 

For  the  parabola,  we  have,  Fig.  92, 

CN^x'  ^p^CB^BN. 

6th.  By  means  of  the  focal  distance  of  the  foot  of  the  normal. 
Subtract  the  value  of  the  intercept  as  found  above,  from 
the  distance  of  the  focus  from  the  centre,  and  the  remainder 
will  be  the  required  distance.     The  parabola  is  the  only 


118.1  ^^0BMAL8  AXD  SUB^ifORMALS.  79 

conic  section  in  which  this  construction  is  of  special  interest. 
For  this  curve  we  have, 


Art.  114a, 

i)iV  =p, 

'^^^^^ 

and,  Art.  86, 

CF=hp, 

and 

CF=I)y-CF=hp;C 
TC=CD',                    "" 

yfl 

\a 

also,  Art.  105, 

C    F    D           N 

Fig.  92. 

adding  gives 

TC  +  CF=  CD  +  DN-  CF; 

or. 

TF=FN; 

also.  Art.  107, 

TF=FP', 

hence  the  points  T,  P,  vV,  are  equi-distant  from  the  focus. 
Hence  with  P  as  a  centre,  and  FF  as  a  radius  describe  an 
arc,  cutting  the  axis  in  the  points  N  and  T ;  the  line  FN 
will  be  the  normal,  and  PT  the  tangent. 

B.  Let  the  given  point  theough  which  the  normal  is 

TO  BE  DRA.WN,  BE  ^TTHIN  OB  WITHOUT  THE  CURVE.   No  general 

method  is  known.  There  are  approximate  methods,  and 
solutions  for  special  points. 

C.  Normal  parallel  to  a  given  line.  Construct  a  tan- 
gent perpendicular  to  the  given  line,  (Art.  126) ;  the  point  of 
tangency  will  be  a  point  of  the  normal,  through  which  point 
erect  a  perpendicular,  and  it  will  be  the  required  normal. 

examples. 

1.  Find  the  equation  of  the  normal  to  the  ellipse  2a;2  +  ^y^  =  40,  at 
a  point  on  the  curve  whose  abscissa  is  2. 

2.  Find  the  equation  of  the  normal  to  the  parabola  y'^  =  4.r,  at  a  point 
whose  abscissa  is  4. 

3.  Find  the  intercepts  of  the  normal  to  the  hyperbola  %x^  —  2y*  =  16, 
at  a  point  whose  ordinate  is  2. 

4.  Find  the  length  of  the  subnormal  to  the  ellipse  J;c*  +  \y'^  =  5,  at  a 
point  whose  ordinate  is  3. 

5.  Required  the  length  of  the  normal  to  the  hyperbola  ^x"^  —  7y* 
=  36,  at  a  point  whose  ordinate  is  4. 

6.  In  the   hyperbola  -q-  —  ^  =  1 ;  required  the   distance  from  the 

centre  of  the  curve  to  the  foot  of  the  normal  drawn  through  the  point  on 
the  curve  whose  ordinate  is  4. 


80 


CONIC  SECTIONS. 


[119. 


7.  Required  the  length  of  the  subnormal  to  the  parabola  x  =  Ay^. 

8.  Required  the  length  of  the  subnormal  in  the  circle  x^  -{-  y'  =  16, 
at  a  point  whose  ordinate  is  2. 

119.  Linear  Equation  to  the  Conic  Sections. — Let 

F'P  =  p=  the  distance  of  any  point  P  from  the  focus  F' ; 

CD  =  X :    then   from  the  figure  we 
have,  as  in  Article  69, 

p'={F'C+CI)y  +  PI?,    . 

=  (ea  +  xf  +  y\ 

Substituting  the  value  of  y\  Eq. 
(oi)  Art.  69,  and  reducing,  we  have 


a'  —  > 
a" 


-x" 


ff  =  ^a''+2eax  +  l^  + 

=  a^+  2eax  +  e^a^; 

.'.  p  =  a  +  ex. 

Similarly,  if  the  pole  be  at  the  focus  F  on  the  positive 
side  of  the  origin,  we  will  find 

p  =  a  —  ex; 
hence,  generally,  p  =  a±ex; 

in  which  p  will  always  be  positive. 

Li  tJie  hyperbola  e  exceeds  unity,  and  x  exceeds  a ;  hence 
in  order  that  p  may  always  appear  to  be  positive  in  the  re- 
sult, we  write  for  this  curve 

p  =  ex±a; 

the  negative  sign  being  used  when  the  pole  is  at  +  c,  and  the 
positive  sign,  when  at  —  c. 

For  the  parabola  we  have,  as  found  in  Article  86, 
FP  =  p  =  x'  +  lp. 

These  equations  being  of  the  first  degree  are  called  the 
linear  equations  of  the  conic  sections. 


130.] 


LINEAR  EQUATION. 


81 


120.  Boscovicli  Definition  of  a  Conic  Section.— The 
linear  equation  of  the  ellipse  may  be  written 


.  =  .(|+x); 


in  which  -   is  constant.      If    a  distance    CL  =  -  be  laid  off 

e  e 

from  the  centre  C,  then  will 
MP=LD  =  LC+  CD 


+  ic; 


G 

^-^ 

"                  ■ 

~^N 

M 

/ 

/^— - 

^ 

k 

T" 

F' 

C 

-Of 

\ 

E 

\ 

/ 

/ 

"  MP-'a 

e 


=  e  =  a    ooni- 


FiG.  94. 


staid  =  the  eccentricity  of  the  ellipse. 

The  abscissa,  x,  of  the  point  may  be  negative,  in  which 
case  the  term  containing  x  wiU  be  negative. 

Similarly,  for  the  hyperbola 


•  =  e{x±l^. 


Make  CL  =— ;  then 
e 

LD  =  MF=CD-  CL 

a 
=  x--; 


and  if  the  directrix,  be  at  the  left  of  (7,  then 

a 


Fig.  95. 


L'D  =  x  + 

e 

and  in  either  case  we  have 

FT*        o 

— — -z=  .^—  e  =  a  constant  =  the  eccentricity 
LD     MP 

of  the  hyperbola. 

For  the  paraMa,  we  have  directly  from  the  definition 


82 


CvmC  SECTIONS. 


[121,  122. 


of  the  curve,  (Art.  82,  and  Fig.  50), 


FP 
CP 


CP 


=  1; 


hence  :  A  conic  section  may  he  defined  as  a  curve  such  that  the 
ratio  of  the  distances  of  any  point  in  it  from  a  fixed  jpoint  and 
fixed  line  is  constant,  and  equal  to  the  eccentricity  of  the 
curve. 

This  is  known  as  Boscovich's  definition.     The  fixed  line 
is  the  directrix. 

121.  The  principles  of  the  preceding  Article  furnish  a 

convenient  method  of  describing 
an  hyperbola  by  a  continuous 
movement.  Let  GL  be  the  di- 
rectrix against  which  moves  the 
triangle  GAE.  Attach  a  string 
at  A,  stretch  it  along  AE,  and 
swing  the  point  E  of  the  string 
around  to  F  and  fasten  it  at  that 
point.  Place  a  pencil  point  at 
P  and  keep  the  string  pressed  against  the  edge  of  the  tri- 
angle while  the  triangle  moves  along  the  directrix.  The  curve 
described  will  be  an  hyperbola ;  for 


Fig.  96. 


PF=PE     AE 


PM 


AG 


=  a  constant 


The  ellipse  may  also  be  constructed  from  the  linear  equa- 
tion, but  the  method  is  somewhat  complex. 

Supplementary  Chords  and  Conjugate  Diameters. 

122.  Supplementary  Chords  are  the   chords  drawn 
from  any  point  of  a  curve  to  the  extremities  of  any  diameter. 

One  of  these  chords  is  supplemen- 
tary in  reference  to  the  other.  Thus, 
if  AA  is  any  diameter,  then  the 
chords  PA  and  PA'  are  supplemen- 
tary in  reference  to  each  other.  The 
Fig.  97.  expressiou  probably  came  from  the 


123.]  SUPPLEMENTARY  CHORDS.  83 

relations  of  these  chords  to  each  other  in  the  circle ;  for  in 
that  curve  the  arc  subtended  by  one  chord  is  the  supplement 
of  that  subtended  bj  the  other. 

123.  Equation  of  Condition  for  Supplementary 
Chords,  in  reference  to  the  Transverse  Axis.— For  the 
ellipse  the  equation  of  the  line  A'P  passing  through  the 
point  A  will  be  of  the  form,  (Art.  38), 

y  —  y  —mix  —x' ) ; 

in  which  y'  =  0,  and  x'  =  —a;  hence  the  equation  becomes 

y  =  m  (x  +  a). 

Similarly,  the  equation  of  the  line  PA  will  be 

y=m'(x  —  a). 

At  the  point  of  intersection  of  these  lines,  both  equations 
will  be  satisfied  for  the  same  values  of  x  and  y.  Multiplying 
the  equations  together,  we  have 

y^  =  m7n'  (x^  —  a^). 

In  order  that  the  point  of  intersection  shall  be  on  the 
ellipse,  the  coordinates  of  that  point  must  satisfy  the  equa- 
tion of  the  ellipse,  which  is,  (Art.  69,  Eq.  (&i) ), 

\P-  — s(ic^—  c?). 

Dividing  this  equation  by  the  preceding  one,  gives 

mm  —  — s; 

which  is  the  required  equation.  The  sign  of  the  product 
being  negative,  it  follows  that  if  one  angle  be  acute  the 
other  will  be  obtuse.  It  will  be  remembered  that  the  an- 
gles are  positive  for  a  left-handed  rotation  measured  from 


84 


CONIC  SECTIONS. 


[134 


For  the  cirde,  we  have  b  =  a,  and  the  equation  of  condi- 
tion becomes 

mm  =  —  1 ; 

^  which  is  the  condition  of  perpendicu- 
larity; hence :  Supplementai'y  chords  in 
a  circle  are  j^erpendicular  to  each  other. 
This  is  another  way  of  stating  the 
principle  given  in  elementary  geometry,  that  an  angle  in- 
scribed in  a  semicircle  is  right. 

For  the  hyperbola,  the  equation  of 
condition  becomes,  by  changing  W  to 


Fig.  98. 


mm  =  —5 
a 


,2» 


Fio.  99. 


which  result   being   positive   shows 
that  both  angles  are  acute  or  both  obtuse. 

In  the  parabola,  the  supplementary  chords  drawn  from  P 
are  PA  drawn  from  P  to  the  vertex  of  the 
curve,  and  PA'  drawn  towards  the  remote 
end  of  the  axis.  Hence  PA'  is  parallel  to 
the  axis  and  makes  with  the  axis  the  angle 
zero;  but  the  angle  which  PA  makes  with 

Fig.  100.  the  axis  is  tan~  — ,  x  and  y  being  the  coordi- 

nates of  the  point  P  in  reference  to  the  vertex  as  an  origin. 


124.  Two    Diameters   are 


Fig.  101. 


Conjugate  ivhen  each  is 
parallel  to  the  tangent  dratvn 
through  the  vertex  of  the 
other.— T\m%  if  PP'  and 
QQ'  are  two  diameters 
such  that  PP'  is  parallel 
to  the  tangent  T'E'  pass- 
ing through  Q  and  QQ' 
parallel  to  the  tangent  at 
P,  then  will  these  diame- 


ters be  conjugate  in  reference  to  each  other. 


125.1  CONJUGATE  DIAMETERS.  85 

125.  Equation  of  Condition  for  conjugate  Diame- 
ters.— For  the  ellipse,  the  equation  of  the  line  CP  passing 
through  the  origin,  will  be  of  the  form 

which  for  the  point  P,  whose  coordinates  are  x'  and  y',  gives 
the  equation  of  condition 

y'  =  m-^  ; 

.'.m,  =  K.  (1) 

X 

The  equation  of  the  tangent  line  E'T  is,  (Art.  100,  Eq. 

{en 

Wx         V 

a-y'        y 

}?x' 
in  which 5—;  is  the  taneent  of  the  obtuse  angle  at  T. 

ary  ^ 


Letting  this  be  represented  by  trii ,  we  have 


h'x 

mi  =  —  ^r> 

a-y 


and  multiplying  (1)  and  (2)  together,  we  have 
msms  =  —  —  , 


(2) 


(3) 


which  is  the  required  equation. 

For  tJie  circle,  a  =  b  =  R,  and  we  have 

mznh  =  —  1 ;  (4) 

and  for  the  hyperbola,  V^  =  —V,  and  we  have 

a^ 
These  values  are  the  same  as  those  found  for  the  condi- 
tions of  supplementary  chords,  (Art.  123),  therefore  we  have 

mm'  =  m^m^ ; 

and  if  m  =  m^  then m'  =  ms;  that  is : 

If  one  diameter  of  an  ellipse  or  hyperbola.,  is  parallel  to 
one  of  two  supplementary  chords,  in  reference  to  the  major 


86 


GONIG  SEGTI0N8. 


[126. 


axis^  the  conjugate  diameter  will  he  parallel  to  the  other 
chord. 

It  also  follows  tliat : 

If  a  jpair  of  supplementary  chords  are  parallel  respectively 
to  a  pair  of  co7ijugate  diameters.,  they  will  also  he  parallel  re- 
spectively to  the  tangents  passing  through  the  vertices  of  those 
diameters. 

The  parabola,  strictly  speaking,  has  no  conjugate  di- 
ameters. 

126.  To  construct  a  tangent  to  a  conic  section  by- 
means  of  conjugate  diameters.  1°.  Let  the  tangent  he 
drawn  through  a  point  P  on  the  curve.      Join  F  with  the 


Fig.  102. 


Fig.  108. 


Fig.  104. 


centre  C  and  through  the  vertex  of  the  major  axis,  draw  the 
chord  A'E  parallel  to  CF  and  join  E  and  A;  the  line  FT, 
parallel  to  EA  will,  according  to  the  preceding  Article,  be 
the  tangent  required.  The  construction  is  the  same  for  the 
circle  and  hyperbola. 

2°.  To  draw  a  tangent  parallel  to  a  given  line.     Let  N  be 


Fig.  105. 


Tig.  106. 


Pig.  lOr. 


the  line ;  draw  a  chord  AE  through  the  principal  vertex  A 
parallel  to  the  line  N,  and  from  E  draw  the  supplementary 


137.] 


CONJUGATE  DIAMETEE8. 


87 


Fig.  108. 


chord  EA' ;  then  from  the  centre  (7 draw  CP  parallel  to  ^'^, 
and  P,  where  CP  intersects  the  curve,  will  be  the  point  of 
tangency,  and  PT  parallel  to  N  will  be  the  required  tan- 
gent. 

For  the  parabola,  let  F  be  the  focus ;  draw  FE  parallel 
to  N,  and  PF  making  the  angle  I^FE 
equal  to  EFX',  then  will  the  point 
P,  when  PF  intersects  the  curve, 
be  the  tangent  point,  and  a  line 
through  P,  parallel  to  K,  will  be  the 
required  tangent.  For,  by  the  con- 
struction, the  angle  T  equals  P,  and 
hence  PP=  PF  as  it  should,  (Art. 
107). 

The  following  construction  corresponds  more  nearly  in 
form  with  that  for  the  other 
conic  sections  than  does  the 
preceding.  Through  the  ver- 
tex of  the  curve  A  draw  a  chord 
AB  parallel  to  the  given  line 
N\  from -P  draw  a  line  BA' 
towards  the  vertex  infinitely 
distant,  it  will  be  parallel  to 
AX.  From  the  centre  draw 
a  line  CP  parallel  to  A'B,  since  it  will  be  a  diameter  it  will 
bisect  the  chord  AB  and  be  parallel  to  AX,  and  the  point  P 
where  it  intersects  the  curve  will  be  the  tangent  point,  and 
PT,  drawn  parallel  to  the  line  N,  will  be  the  required  tan- 
gent.    See  also  Article  143. 

127.  Equation  to  the  Ellipse  referred  to  Oblique 
Axes,  the  origin  being  at  the  centre.  The  equations  for 
changing  from  rectangular  to  oblique  axes,  the  origin  being 
the  same,  are,  (Art.  52,  Eqs.  (e)). 


Fig.  109. 


x=zx'  cosa  +  y'cos/?; 
y  =x'  sin  a  +  y'  sin  /?. 


Substituting  these  values  in  the  equation  of  the  ellipse 


88  CONIC  SECTIONS.  [128. 

referred  to  its  axes, 

and  arranging  the  result,  gives 

a^sin^yS  [ 
+  h^  cos' 


ft  I    ,2+  «'  sin^^  )  ^,^+  a^sinn'sin/?  )  g^V'  =^2^2  . 
y^  j  ^    +6^cos-r<r  j'      +  S^COSO'COS/J  j        ^  ' 


whicli  is  the  required  equation,  in  which  a  is  the  angle  be- 
tween the  axis  x'  and  the  major  axis  of  the  ellipse,  and  fd  the 
angle  between  y'  and  the  same  axis.  For  the  hyperbola, 
change  6^  to  —  h^. 

128.  Ellipse  referred  to  Conjugate  Diameters. — ^In 
order  that  the  new  coordinate  axes  shall  coincide  with  conju- 
gate diameters,  they  must  be  subjected  to  the  condition  given 
in  Article  125,  or 


m.mn  =  — 


h- 


which  becomes 

tan  a  tan  /?  = ^ ; 

which  may  be  reduced  to 

a?  sin  a  sin  /3  +  Jf  cos  oc  cos  ft  =  0. 

This  condition  causes  the  coefficient  of  x'y'  in  the  equa- 
tion of  the  preceding  Article,  to  disappear,  and  that  equation 
becomes 

{a"  sin*  ft  +  V'  cos^  ft)  y'^  +  (a'  sin^  a  +  h"  cos^  a)x"'  =  a'V^. 

In  this  equation,  if  x  =  0,  we  have 

,  ah 

y  = 


Vo?  sin"  ft  +  ?/  cosV  ' 

which  is  the  intercept  on  the  axis  of  y'.    Let  this  be  repre- 
sented by  b'.     Similarly,  the  intercept  on  x  will  be 

ab 
Vd  sin  a  +  tr  cos  «  ^    '' 


X  = 


139,  130.]  CONJUGATE  DIAMETERS.  '  89 

These   values   reduce   the  preceding  equation  to,  after 
.    dropping  the  accents  fi'om  the  variables, 

9  9 

X  IJ' 

^2  +  p  =  1 ;  or  a '  -if  +  h '  -,i"  =  a '-h'  \  (1) 

which  is  the  equation  of  the  ellipse  referred  to  conjugate  di- 
ameters. 

For  the  circle,  a'  =  b'  —  E,  and  equation  (1)  becomes 

|,+  g=l,  (2) 

which  is  the  same  as  that  previously  found  for  rectangular 
axes,  as  it  should  be,  since  the  conjugate  diameters,  in  this 
curve,  are  at  right  angles  with  each  other. 

For  the  hyperbola,  b'^  =  —b'^,  and  equation  (1)  becomes 

^.-f;=l.  (3) 

These  equations  are  of  the  same  form  as  those  deduced 
for  rectangular  axes. 

129.  Discussion  of  equation  (1). — Solving  for  x  gives 

x=  ±jWb''-y^; 

which  shows  that  for  every  value  of  y  less  than  b',  there  are 
two  equal  and  opposite  values  for  x ;  hence  the  curve  is 
<Miqudy  symmetrical  in  reference  to  the  axis  of  y.  In  the 
same  manner  we  find  a  like  symmetry  in  respect  to  x.  Since 
the  axis  of  x  can  be  made  to  coincide  with  any  diameter,  it 
follows  that  every  diameter  bisects  a  system  of  chords  parallel  to 
its  conjugrtte. 

This  equation  may  be  discussed  in  other  respects  the 
same  as  for  rectangular  axes.  Equation  (3)  gives  corre- 
sponding results. 

130.  Equation  of  the  Tangent  referred  to  conjugate 


90 


CONIC  SECTIONS. 


[181. 


diameters.     The  equation  to  the  curve  being  of  the  same 

form  as  for  rectangular  axes, 
the  equation  to  the  tangent 
will  be  of  the  same  form. 
We  have  therefore  only  to 
substitute  a'  for  a,  and  h' 
for  h  in  the  equations  pre- 
viously found.  Let  P  be  the 
point  of  tangency,  CD  =  x\ 
PD  =  y' ;  then  will  the  equa- 


FiG.  110. 


tion  of  the  tangent  PThe,  (see  Art.  100), 
for  the  dlipse, 


X  X      y'y 


For  the  hyperbola,  (see  Art.  102), 


XX 


yy  -A 


Fia.  111. 


For  the  intercept  CT  make  y  =  0,  and 
we  have,  for  the  ellipse,  circle,  and  hyper- 
bola, 


CT  =  —  . 

X 


In  a  similar  manner  the  subtangent  DT  may  be  found. 

[For  a  shorter  course  omit  to  Article  167.] 

131.  Transform  the  equation  to  the  Ellipse  from 
conjugate  diameters  to  its  axes.  — The  formulas  for 
passing  from  oblique  to  rectangular  axes,  the  origin  remain- 
ing the  same,  are,  (Art.  53), 


,  _  icsin  fi—y  cos ^ ^ 


y 


ycosa  —  xsma 


sin(/^-a)     '  ^  sin(/i-«')      * 

These  values  substituted  in  the  equation  to  the  ellipse 
referred  to  conjugate  diameters,  which  is,  (Art.  128,  Eq.  (1)), 

gives 


131.]  CONJUGATE  DIAMETERS.  91 

(a ' ^  cos"-  o(  +  h''-  cos'^ yS)  y^  \ 
+  (a '  2 sin- a+h'-  sin' /i).r-    I  =  « ' 2 6' 2  g^^^s  ^^  _  ^^^ 
—  2  (rt ' 'sin  «  cos  a  +  h"^  sin /i cos  /:^)  xy  ) 

TMs  equation  must  be  reduced  to  the  form 
a^y-  +  Ir.i^  —  d-Jr, 

previously  found  for  the  equation  to  the  ellipse  referred  to 
its  axes,  (Art.  69,  Eq.  (r/j)). 

Comparing  the  two  preceding  equations,  we  find  : 

a'^  cos^o- +  6'^cos'/S  =  a-;  (1) 

a'2sin2«  +  6'2sinV  =  ?^;  (2) 

a'^  sinacosa  +  Z)''-sin/?cos /5  =  0  ;  (3) 

a'H'^sm^  {^-a)=a:W.  (4) 

These  are  the  equations  of  condition,  and  from  them  we 
deduce  the  following  results  :  Adding  the  first  and  second 
gives 

or  4a'2  +  4&'2  =  4a-^  +  4&2;  (5) 

that  is :  The  sum  of  the  squares  of  any  pair  of  conjugate  di- 
ameters of  an  ellipse  is  constant,  and  equals  the  sum  of  the 
squares  of  the  axes. 

When  the  axes  and  one  of  the  conjugate  diameters  is 
given,  the  other  conjugate  diameter  may  be  found  from  the 
preceding  equation. 

Changing  Ir  to  —  &',  and  &'  ^  to  —  6'  ^  gives 

that  is :  For  the  h37perbola  the  dijference  of  the  squares 
of  any  two  conjugate  diameters^  equals  the  difference  of  the 
squares  of  the  axes. 

Because  the  aoxs  are  conjugate,  we  must  have,  (Art.  128), 

o^  sin  a  sin  ft  =  —W  cos  a  cos  /? ; 

.".  tan  or  = 5  cot  /?,  (7) 


92 


CONIC  SECTIONS. 


[181. 


from  wliicli  one  of  the  angles  may  be  found  when  the  other 
angle  and  the  axes  are  given. 

It  appears  that  if  one  of  these  angles  is  acute  the  other 
will  be  obtuse.     From  equation  (3)  we  have 

a'  ^  sin  a  cos  a  =  —  b'"  sin  p  cos  /3  ; 

which  combined  with  equation  (5)  by  eliminating  h',  gives 


a?  +  b' 


sin  a  cos  a 
sin  p  cos  p 


(8) 


from  which  one  of  the  conjugate  diameters  may  be  found 
when  the  axes  are  given,  and  the  angles  a  and  p  have  been 
found.  The  other  axis  may  be  found  in  a  similar  manner, 
or  the  value  of  a  may  be  substituted  in  equation  (5),  and  b' 
deduced  from  the  result. 
From  equation  (4)  we  find 

^^  ia'h' Bin  (P- a)  =  4ab;  (9) 

the  second  member  of  which 
represents  the  area  of  the 
rectangle  constructed  on 
the  axes  of  the  ellipse.  In 
the  figure 


Fio.  112. 

PCA  =  a,  MCA 


P\  .\MCP  =  P  -a. 


Construct  the  parallelogram  EFGH,  having  its  sides  tan- 
gent to  the  ellipse  at  the  extremities  of  the  conjugate 
diameters  PP'  and  MM' ;  then  will  3ICPG  be  one-fourth  of 
the  circumscribed  parallelogram.  Since  the  sum  of  the 
angles  of  a  parallelogram  equals  four  right  angles,  we  have 

GMC  +  3ICP  =  180° ; 

.'.  sin  GMC  =  sin  (180°  -  MCP)  =  sin  {P  -  a), 
and       {GM=  a' )  sin.  (p  —  a)  =  the  perpendicular  GD, 
and         a'h'  sin  {P  -  a)  =  MC.  DG  =  area  MCPG ; 
.-.  Aa'b'  sin  {p  -  a)  =  EFGH; 


132,  183.] 


THE  PARABOLA. 


93 


hence  :  The  area  of  a  parallelogram  circu7nsGriMng  an  ellipse 
in  which  the  sides  are  tangent  to  the  ellipse  at  the  vertices  of  a 
pair  of  conjugate  dianieters,  is  constant  and  equal  to  the 
rectangle  constructed  on  the  axes. 

For  tJie  hyperbola  we  substitute 
V—  ib  for  b,  and  a/—  16'  for  b',  and 
obtain  precisely  the  same  result,  since 
V—  1  drops  from  the  equation ;  hence 
the  preceding  conclusion  is  true  for 
the  hyperbola. 

•'  ^  Fig.  113. 


EXAMPLE. 

If  the  major  axis  of  an  ellipse  is  20  aiid  the  minor  axis  is  12,  iind  the 
conjugate  diameters  and  the  angle  fJ,  when  a  is  45° ;  deduce  the  area  of 
the  circumscribed  parallelogram  whose  sides  are  parallel  to  the  conjugate 
diameters  and  show  that  it  equals  the  area  of  the  rectangle  on  the  axes. 

132.  The  Parabola  referred  to 
Oblique  Axes. — Substituting  the  values 
of  X  and  y  from  equations  (d)  Article 
62,  in  the  rectangular  equation  of  the 
curve, 

y^  =  2px, 


gives,  dropping  the  accents, 


Fig.  114. 


sin- f:i .if  +  sin^ a.  x"^  +  2 ^in a s,in^.xy  \ 

+  2{n  sin  /5  — ^  cos  p)y  +  2  {n  sin  a—  p  cos  a)x\-—0, 
+  n^  —  2pm  ) 

which  is  the  required  equation. 

133.  Parabola  referred  to  a  Diameter  and  a  Tangent 
at  its  vertex. — The  equation  for  this 
case  can  be  deduced  directly  from  the 
preceding  one.  Since  the  new  origin 
will  be  on  the  curve,  the  co  irdinates 
of  which  in  reference  to  the  vertex  are 
rn  and  n,  we  have  from  the  equation 
of  the  curve 


n^  =  2pm. 


Fig.  115. 


94  CONIG  SECTIONS.  [ISa 

Since  the  new  axis  of  x  is  to  be  parallel  to  the  axis  of  the 
curve,  we  have, 

«  =  0 ;  .•.  sin  «  =  0,  and  cos  a  =  l. 
Because  OZ'  is  a  tangent,  we  have,  (Art.  103), 
yn  —  p  {x  +  m), 

or  y—^{x  +  m)'. 


n 


,      ^      sin  /?     p 
cos  p     n 

or  wsin/?— ^cosyS  =  0. 

These  several  values  substituted  in  the  equation  of  the  pre- 
ceding Article,  reduces  it  to 

^  ~  sin^/J^^ 
or,  if  2p'  be  the  coefficient  of  x,  we  have 

if  =  2p'x, 

which  is  the  required  equation,  and  is  of  the  same  form  as 
when  the  curve  is  referred  to  its  axis  and  a  tangent  at  its 
vertex. 

Discussion  of  the  equation  y^  =  Ip'x. 

1°.  For  every  positive  value  of  x  we  have 


?/  =  ±  V2p'x ; 

that  is :  The  ordinates  paraUd  to  the  tangent  at  the  vertex  of  any 
diameter,  are  bisected  by  that  diameter;  or  the  curve  is  oUiqvdy 
symmetrical  in  reference  to  every  diameter. 

2°.  For  negative  values  of  r,  y  is  imaginary,  hence  :  The 
curve  does  not  extend  on  the  negative  side  of  the  tangent. 

3".  For  two  points  on  the  curve,  whose  abscissas  are  re- 
spectively Xi  and  a?2,  we  have 


134.] 
or 


THE  PARABOLA. 


95 


yi  '  y-2 


X.-,', 


that  is:  The  squares  of  the  ordinates  to  any  diameter  are  as 
tJieir  corresponding  abscissas. 

4°.  The  equation  referred  to  a  diameter  and  a  tangent  at 
its  vertex  being  of  tlie  same  form  as  the  rectangular  equa- 
tion previously  found,  it  follows  that  the  equation  of  the 
tangent  line  will  be  of  the  same  form,  hence,  if  the  point  of 
tangency  be  x'y\  we  have,  for  the  equation  of  the  tangent. 


yy'  =  p  {x  +  x)  \ 


and  if  y  ~0,  we  have 


tV     -"■    vO       y 


that  is  :  The  subtangent  to  any  diameter  is  bisected  at  the  vertex 
of  that  diameter.  Also  the  two  tangents  at  the  extremities  of  any 
ordinate  intersect  on  the  axis  of  x. 

134.  "Value  of  h    .  ,  -.    Let  F"  be  the  focus  ;  A '  the  ver- 
*"  sin^  p 

tex  of  the  diameter  A'F' ;  x'  =  AD,  the  abscissa  of  ^' ;  y' 
=  A  'D,  its  ordinate  ;  and  TA  ',  a  tangent.  According  to  Ar- 
ticle 103,  we  find, 

tan^'rj^  =  tajiyS  =  ^; 

y 

sin'  /?    _  1^  _   1^  _   P  . 


hence,  solving  for  sin^  ^,  gives 


sin-y^  = 


Fig.  116. 


P 


2x'  +  p ' 


3  -^fi  =  ^'  +  IP- 

sm  p 


But  TA  =  AD=x',  (Art.   104);  AF=lp;   TF=FA\ 
(Ari  107,  Eq.  (6)) ; 

.-.  x+lp=  TF=FA'  =  h^^'y 


96  CONIC  SECTIONS.  [135-137. 

that  is :  I'he  distance  of  the  vertex  of  any  diameter  of  a  parah- 
olafrom  the  focus  equals  the  quotient  found  hy  dimdiiig  the 
distance  of  the  focus  from  the,  principal  vertex  hy  the  square  of  the 
sine  of  the  inclination  of  t/ie  tangent  at  tJie  vertex  of  the  diameter. 

135.  Other  relations. — Let  EG  be  the  directrix,  and 
PP'a  chord  through  the  focus,  parallel  to  the  tangent  A'  T. 
Then,  by  the  construction,  TF  =A'F',  and,  according  to  Ar- 
ticle 83,  P^ '=:  ^'i^ ; 

.-.  TF=FA'  =  EA'  =A'F'=h-J^. 

136.  Any  double  focal  ordinate  equals  2p-f-sin-y5; 
in  which  /?  is  the  inclination  of  the  ordinate  to  the  axis  of 
the  curve.     For,  in  the  equation 


y- 


_     %> 


sinV^ 


substitute  x=  OR,  (Fig.  115),  =  |  .  ^    ,  and  we  find 

y=  ±J^^RP', or  RP. 
^  sin^  fi  ' 

But  PP'  =  2y; 

•*  ~sin2/?' 

which  was    to  be  proved.      This  value  is  also  called  the 
parameter  to  any  diameter.  Article  145. 

137.  Problem.  Find  the  point  of  intersection  of  a  tan- 
gent to  tlie  parabola  with  a  perpendicular  upon  the  tangent  from 
the  focus. 

Let  BP  be  the  tangent,  P  the  point  of  tan- 
"p  gency  whose  coordinates  are  x,  y',  and  FB  the 

BJ^  perpendicular  from  F  upon  BP.     The  equation 

to  the  tangjent  will  be,  (Art.  103), 

Fi«-  "7.  y=  ^,(x  +  x'); 

and  of  the  line  FB  passing  through  the  point  dp,  0)  and 
perpendicular  to  BP  will  be,  (Art.  46), 

2/  =  -  1^  (^  -  hp)' 


138,  139.] 


TANGENTS  TO   THE  PARABOLA. 


97 


Combining  these  equations  gives 


0, 


y  =  W 


The  equation  a;  =  0,  shows  that  the  intersection  will  be 
on  the  tangent  passing  through  the  principal  vertex;  and 
conversely : 

Perpendiculars  drawn  to  the.  focal  radii  at  their  intersec- 
tion with  the  tangent  at  the  principal  vertex,  will  he  tangent  to 
the  parabola. 

The  condition  y  —  l^y'  furnishes  an  easy  mode  of  drawing 
a  tangent  to  the  curve  ;  for  the  distance  AB  equals  one-half 
the  ordinate  of  the  point  of  tangency.  This  is  substantially 
the  same  as  in  Article  108,  2°. 


Geometrical  Construction  of  the  Parabola. 

138.  Given  thefocns,  and  the  tangent  at  the  prin/iipal  vertex 
to  construct  the  parabola.  Let  F  be  the 
focus;  draw  radial  lines  F\,  F%  etc.,  and 
at  their  intersection  with  the  tangent, 
1,  2,  etc.,  draw  perpendiculars  to  the 
radial  lines.  The  curve  drawn  tangent 
to  the  successive  lines  will  be  a  parab- 
ola. The  greater  the  number  of  radial 
lines,  the  more  accurately  can  the  curve  be 
drawn. 

Fig.  118. 

139.  To  constrvxit  a  parabola  by  bisecting  subtangents. — Let 
TG  and  TB  be  the  tangents.  Join  G  and  B  and  bisect  at 
A.  Draw  TA  and  bisect  at  E; 
then  will  .£'  be  a  point  in  the 
curve.  For  TA  drawn  from  the 
intersection  of  the  tangents  to 
the  middle  of  the  chord  GB,  is 
a  diameter,  and  TA  is  a  sub- 
tangent  on  that  diameter,  and 
will  be  bisected  by  the  curve 
at  E.  Join  E  and  G,  bisect  at 
G,  draw  FG  and  bisect  at  H, 
and  H  will  be  a  point  on  the 

7 


Fie.  119. 


98 


CONIC  SECTTONS. 


[140-143. 


curve.     In  a  similar  maimer  any  number  of  points  may  be 
found. 

140.  By  means  of  enveloping  tangents. — Divide  the  tan- 
gents into  the  same  number  of  equal 
parts,  and  number  them  in  reverse 
order  as  shown  in  the  figure,  and 
join  the  corresponding  numbers  by 
straight  lines.  The  curve  will  be  tan- 
gent to  the  successive  lines.  For  the 
intersection  of  the  corresponding  tan- 
gents will  be  on  a  right  line  and  the 
tangents  be  equal  to  each  other ;  hence 
the  subtangents  of  the  corresponding 
pair  of  tangents  will  be  equal  to  each  other. 

141.  By  means  of  bisecting  tan^ 
gents. — Let  TA  and  TB  be  two  tan- 
gents. Bisect  them  in  C  and  /,  and 
bisect  IC  in  E;  then  will  ^  be  a 
point  in  the  curve,  Art.  139.  Bi- 
sect GE  at  F,  and  CA  at  H,  then 
will  the  middle  point  of  FH  at 
e7  be  a  point  in  the  curve,  and  so 
on. 


Pig.  120. 


FiQ.  121. 


142.  By  means  of  intersections. — 
Let  TO  and  TD  be  the  tangents, 
c  Draw  TB  to  the  middle 
a  point  of  CD,  and  bisect 
it  at  4;  this  will  be  one 
point  of  the  curve.  Di- 
vide T4:  and  TC  into  the 
same  number  of  equal 
parts.  Draw  lines  from 
C  to  the  points  of  division 
in  T4,  and  through  the 
points  d,  e,  f,  etc.  lines 
parallel  to  TB ;  then  will 
the  intersection  of  Cm^  and/c  at  3  be  one  point  in  the  curve ; 
Owj  and  eb  at  2  be  another  point,  and  so  on.     For  the 


Fio.  122. 


143-145.] 


TANaENTS  TO   THE  PARABOLA. 


99 


distances  from  d,  e,  etc.  will  vary  as  the  squares  of  Cd^ 
Ce,  etc. 

Points  may  also  be  found  by  dividing  4:B  into  the  same 
number  of  equal  parts  as  in  T4:  and  drawing  lines  (7«i  etc.  to 
intersect  lines  Drrii  etc. ;  the  points  of  intersection  will  be 
points  in  the  curve. 

143.  To  draio  a  ta/ngent parallel  to  a  given  Utic. — Let  JV  be 
the  given  line.  Draw  two  chords 
AB  and  BE  parallel  to  A\  and 
bisect  them  in  C  and  F;  then  will 
CFO  drawn  through  F  and  0  be 
a  diameter.  Through  the  vertex 
0  of  this  diameter  draw  07^  par- 
allel to  JV,  then  will  0  7"  be  the  fig.  123. 
tangent  required,  (Art.  133, 1°). 

144.  To  Jind  the  axis  of  a  parabola. — Draw 
any  two  parallel  chords  and  bisect  them  by  a 
line  OF '.  This  line  will  be  a  diameter.  Draw 
chords  perpendicular  to  OF',  and  bisect 
them  by  the  line  AX;  then  will  AX  be  tke 
axis  of  the  parabola. 

Fig.  124. 


Of  Parameters. 

145.  In  the  Ellipse,  (Circle),  and  Hyperbola,  the 
Parameter  in  respect  to  any  diameter  is  defined  to  be  tJm 
third  proportional  of  that  diameter  to  its  conjugate.  Hence  the 
parameter  to  the  diameter  2a'  will  be 

2a'  :  26'  : :  2b'  :  parameter ; 
{2b'f_2h'^ 


.'.  parameter 


2a 


a 


2^ 


In  reference  to  the  major  axis,  this  becomes  — ,  which  is  the 

value  of  the  latus  rectum,  (Art.  94). 

Similarly,  the  parameter  to  the  diameter  2V  is 

26'  • 


100 


CONIG  SECTION'S. 


[146,  147. 


146.  In  the  Parabola  the  parameter  in  respect  to 

ANY  DIAMETER  ^s  defined  to  be  the  third  projjortional  of  any  ab- 
scissa to  its  corresponding  ordinate,  the  ordinate  being  paxallel 
to  the  tangent  at  the  vertex  of  that  diameter ;  lience 

parameter  =  ^=   .  ^  ,.,  (Art.  133), 

uL'  SlU    /^ 

which  equals  the  double  ordinate  through  the  focus,  (Art. 
136). 

If  the  diameter  is  the  axis  of  the  curve,  this  value  be- 
comes, (since  /S  =  90''), 

which  is  tJie  lattLs  rectum,  (Art.  95). 

Pole  and  Polar. 

147.  The  Polar  of  any  point  in  respect  to  a  conic  sec- 
tion, is  the  locus  of  the  intersection  of  the  two  tangents 
drawn  at  the  extremities  of  any  chord  passing  through  the 
point.  It  will  be  shown  in  Article  152,  that  this  locus  is  a 
straight  line.     Thus,  if  P  be  any  point  in  a  conic  section. 


Fig.  125. 


MM'  and  NN'  two  chords  drawn  through  this  point,  then 
will  the  point  T,  the  intersection  of  the  tangents  TM  and 
TM',  be  one  point  in  the  polar,  and  T\  the  intersection  of 
T'N  and  T'N',  another  point,  and  the  line  TT  passing 
through  these  points  will  be  the  polar  required.  The  chords 
MM'  and  NN'  are  called  chords  of  contact,  in  reference 
to  the  tangents  drawn  from  their  extremities. 


148-150.] 


POLE  AND  POLAR. 


101 


148.  The  Pole  of  any  right  line,  with  respect  to  a 
conic  section,  is  the  intersection  of  the  chords  of  contact  in 
reference  to  points  on  the  polar.  Thus,  P  is  the  pole  in 
reference  to  the  polar  TT '.  A  polar  point,  in  reference  to 
the  pole,  is  the  point  where  the  diameter  through  the  pole 
intersects  the  polar.  The  pok  and  polar  are  reciprocal 
terms ;  neither  has  any  signification  without  the  other. 

149.  If  the  pole  P  is  without  the  curve,  the  polar 
will  be  the  chord  of  contact.  For,  if  any  secant  PE  be 
drawn,  the  part  EE'  will  be 
the  chord  drawn  through  the 
pole.  The  intersection  of  the 
tangents  passing  through  E 
and  E'  (but  which  are  not 
shown  in  the  figure)  will  be  a 
point  on  the  polar.  If  now 
the  secant  be  turned  about  P, 
the  points  E  and  E '  will  approach  T,  and  also  the  intersec- 
tions of  the  tangents  will  continually  approach  the  same 
point,  and  when  E  and  E'  are  consecutive  to 
T,  the  tangents  will  intersect  at  that  point. 
Hence,  T  is  one  point.  For  the  same  reason, 
T'  is  another  point.  If  the  pole  is  on  the  curve, 
the  polar  will  be  tangent  to  the  curve  at  that 
point. 

150.  Equation  to  the  Chord  of  Con- 
tact.— First  consider  the  ellipse.  The  equation  .of  the  tan- 
gent referred  to  conjugate  diameters,  the  point  of  tangency 
M  being  x'  y',  (Fig.  125),  will  be,  (Art.  130), 


Fig.  129. 


XX 


yy 


1; 


and  for  the  point  x"y",  or  M\ 


XX 


+  o=i. 


But  these  tangents  will  intersect  in  some  point  as  T,  which 
denote  by  Xxy\\   hence  the  coordinates  of  this  point  must 


102  come  SECTIONS.  [150. 

satisfy  the  equations  of  both  lines,  and  we  have  ih£  equations 
of  condition, 

^1^-1.  ^14.^1  =  1 

By  means  of  these  equations  the  point  of  intersection 
ociyi  may  be  found  when  the  curve  and  the  points  of  contact 
are  known.  Having  found  this  point,  let  x"  and  y"  or  a?" 
and  y'  in  the  last  equations  be  changed  to  general  variables, 
and  we  have 

which,  being  an  equation  of  the  first  degree,  is  the  equation 
of  a  right  line.  This  line  must  pass  through  the  point  x"  y" 
because  these  coordinates  satisfy  the  equation  of  the  line,  as 
shown  above  ;  and,  for  the  same  reason,  it  will  also  pass 
through  x'y' ;  hence  it  is  called  the  equation  to  the  chord  of 
contact. 

For  the  circle,  make  a'  =  V  =  B,  and  we  have, 

for  the  equation  of  the  chord  of  contact  for  the  circle. 

For  the  hyperbola,  we  have,  for  the  equation  of  the  chord 
of  contact, 

a'^        J' 2       -L-  W 

For  the  parabola,  we  would  find 

y,y=p'{x  +  Xi).  (4) 

If  all  these  curves,  except  the  parabola,  are  referred  to 
their  axes,  we  have  a'  =  a,  h'  —  b,  and  in  the  parabola,  if  re- 
ferred to  its  axis  and  a  tangent  at  its  vertex,  p'  =  p.  The 
preceding  equations,  therefore,  remain  of  the  same   form 


151.]  POLE  AND  POLAR.  103 

whetlier  the  ciirres  are  referred  to  their  axes  or  conjugate 
diameters. 

151.  Remark. — The  changes  from  variables  to  con- 
stants, and  from  constants  to  variables,  may  appear  to  the 
beginner  arbitrary  and  meaningless.  We  will,  therefore, 
add  a  few  simple  illustrations.  Take  a  true  numerical  ex- 
pression, as 

9  .  2  +  4  .  3  =  30  ; 

and  changing  2  to  x,  and  3  to  y,  we  have 

9x  +  42/  =.  30 ; 

which  is  the  equation  of  a  right  line.  This  line  wiU  pass 
through  the  point  (2,  3).  The  above  numerical  equation  is 
of  the  form 

a^'-^i  +  y'yi  =  30. 

If  now,  in  the  last  equation,  we  make 

x  =  X,  and  y'  =  y , 
we  have  XiX  +  y^y  =  30 ; 

and  making  jTi  =  2,  yi  =  3  ; 

it  becomes  2a;  +  3^/  =  30 ; 

which  is  the  equation  of  a  right  line  passing  through  the 
point  (9,  4),  since  these  values  satisfy  the  equation. 
If  we  have  the  two  numerical  equations 

9.1-4.2  =  1; 

9  .  2  -  4  .  4i  =  1 ; 

they  may  be  represented  by 

Xxx'  -  yiy'  =  1 ;         Xix"  -  y^y"  =  1 ; 

and  if  x  and  y\  or  x"  and  y"  be  made  general  variables,  we 
have 

XiX  +yiy  =  l; 

which  is  the  equation  of  a  right  line  passing  through  the 
points  (1,  —  2)  and  (2,  —  4^),  for  the  coordinates  of  both  these 
points  satisfy  the  equation. 


104  CONIC  SECTIONS.  [152. 

In  these  equations  the  values  of  Xx  and  y^  have  been 
assumed  arbitrarily ;  but  in  tlie  equation  to  tlie  chord  of  con- 
tact, given  in  the  preceding  Article,  they  are  determined 
by  means  of  a  certain  condition  (being  the  co  rdinates  of  the 
point  of  intersection  of  certain  tangents),  and  this  condition 
must  be  realized  so  long  as  x^  and  y^  remain  in  the  equation. 

152.  Equation  to  the  Polar. — In  regard  to  the  ellipse, 
we  found,  in  Article  150,  that  the  equation  to  the  chord  of 
contact  is 

a''  +   J'2  -  -L-  W 

If  this  line  passes  through  a  fixed  point  P,  whose  coordi- 
nates are  x^,  ^2*  we  will  have  the  equation  of  condition 

^/2  "•"    ;.'2  —  ■^>  \^^ 


If  now  Xi  and  yi ,  the  coordinates  of  the  intersection  of  the 
tangents,  be  changed  to  general  variables,  we  have 

which  will  be  the  equation  to  the  locus  of  the  intersection  of 
the  tangents  drawn  from  the  extremities  of  any  chord  pass- 
ing through  the  point  P.  It  is  the  equation  of  a  right  line, 
and  is  the  equation  to  the  polar.  It  is  of  the  form  of  the 
equation  to  the  tangent  line. 

For  the  circle,  the  equation  to  the  polar  becomes 

and  for  the  hyperbola, 

x^     y,y  _^  ,^s 

For  the  parabola,  when  the  curve  is  referred  to  a  diam- 
eter and  a  tangent  at  its  vertex,  we  would  find 


153, 154.]  POLE  AND  POLAR.  105 

y,y  =p'{x+x.;):  (6) 

in  whicli  p  —  p,  if  the  diameter  is  the  axis. 

If  all  these  curves,  except  the  parabola,  are  referred  to 
their  axes,  we  have 

a  =  o,  and  h'  =  h. 

The  preceding  equations  show  that  the  polar  of  any  point 
in  respect  to  any  conic  section  is  a  right  line. 

153.  Direction  of  the  Polar. — Comparing  the  equations 
of  the  preceding  Article  with  those  of  Article  130,  shows 
that : 

The  polar  to  any  point,  in  respect  to  a  conic  section,  is  parallel 
to  the  system  of  chords  bisected  by  that  diameter  which  passes 
through  tJie  pole. 

From  this  principle  it  follows  directly : 

1°.  That  for  a  pole  on  the  axis,  the  polar  ivill  be  perpendicular 
to  that  axis. 

2°.  For  any  point,  the  polar  ivill  he  parallel  to  the  tangent  at 
the  extremity  of  the  diameter  passing  through  the  point. 

154.  Polar  of  Special  Points.  If  th£  pole  be  at  the 
centre,  we  have,  for  all  but  the  parabola, 

X2  =  0,  2/2  =  0; 

which  substituted  in  equations  (3)  and  (5)  of  Art.  152,  give 

X       0  2/  _  1  _ 

'^^'^obr^~o~'^' 

which  is  the  equation  of  a  right  line  at  infinity,  whose  direc- 
tion is  indeterminate  ;  hence  :  The  polar  of  the  centre  is  any 
right  lin£  at  infinity. 

If  the  pole  be  at  the  vertex  of  the  major  axis,  we  have,  for  all 
but  the  parabola, 

X2=T  a,  2/2=0; 

.'.  x  =  0.y  ±n, 

which  is  the  equation  of  a  line  parallel  to  the  axis  of  y,  and 
at  a  distance  from  it  equal  to  a.     Hence,  as  the  pole  passes 


106  CONIG  SECTIONS.  [155. 

from  the  rertex  of  the  major  axis  to  the  centre,  the  polar 
passes  from  a  distance  a  from  the  centre,  to  an  infinite  dis- 
tance. 

155.  Polar  to  the  Focus. — ^For  the  ellipse  and  hyperbola^ 
we  have 

rcg  =  ±  ae,  2/2  -  0, 

which  in  equations  (3)  and  (5)  of  Article  152,  give 

x  =  0.y±-; 

which  is  the  equation  to  a  line  parallel  to  the  asis  of  y^ 
hence  :  The  polar  to  either  focus  is  a  perpendicular  to  the  major 
axis  and  cuts  that  axis  at  a  distance  from  the  centre  equal  to 
a-T-e,  measured  on  the  same  side  as  the  focus. 

The  polar  to  the  focus  is,  therefore,  the  directrix^  Art.  120. 

For  the  parabola,  we  have  for  the  focus, 

^■2  =  Ip,  1/2  =  0; 

which  in  equation  (6),  Article  152,  gives  by  reduction, 
x  =  0.y  -Ip; 

which  is  the  equation  of  the  directrix.  Hence  :  The  polar  to 
the  focus  of  any  conic  section  is  the  directrix  of  the  curve. 

Tangents  at  the  extremities  of  a  focal  chord  are  called 
focal  tangents ;  hence  the  two  preceding  conclusions  may  be 
stated  as  follows : 

The  locus  of  the  intersection  of  the  focal  tangents  is  the  direo- 
trix. 

EXAMPLES. 

1.  Required  the  equation  to  the  polar  of  the  point  ar^  =  2,  y^  =  1,  in 
respect  to  the  ellipse  2y*  +  3a;*  =  27. 

2.  Find  the  polar  of  the  point  iCx  =  —  1,  y2  =  0,  in  respect  to  the  circle 
a!«  +  y«  =  9. 

3.  Find  the  equation  to  the  polar  of  the  point  a;2  =  1,  yj  =  0,  in  re- 
spect to  the  hyperbola  3a;*  —  t/*  =  12. 

4.  Required  the  polar  of  the  point  X2  =  3,  2/2  —  0,  in  respect  to  the 
parabola  y'  =  6a;. 

5.  Given  the  polar  y  =  2x  +  8,  to  find  the  pole  in  respect  to  the  circle 
x'  +y^  =  9. 


156, 157.]      THE  HTPEEBOLA  AND  ITS  ASYMPTOTES.  107 

6.  Given  the  polar  y  ~  z  +  2^  to  find  the  pole  in  respect  to  the  pa^ 
rabola  i/^  =  4a;. 

The  Hyperlola  and  its  Asymptotes. 

156.  Definition.  An  Asymptote  to  a  curve  is  a  line 
towards  wliich  the  tangent  continually  approaches  as  the 
tangent  point  moves  away  from  the  origin  indefinitely. 

According  to  this  definition,  the  curve  must  have  an  infi- 
nite branch  ;  hence,  neither  the  circle  nor  the  ellipse  can 
have  an  asymptote. 

The  parabola  has  an  infinite  branch  ;  but  it  is  shown  in 
Article  104,  that  the  intercepts  of  the  tangent  increase  in- 
definitely as  x  increases  indefinitely  ;  hence  the  tangent  does 
not  approach  a  definite  position  as  a  limit. 

But  in  the  hyperbola,  intercepts  on  both  axes  are  finite 
when  the  abscissa  of  the  point  of  tangency,  x,  is  infinite. 
If  the  origin  of  coordinates  be  at  the  centre  of  the  hyper- 
bola, the  intercepts  will  be  zero,  when  x  =  oo,  (Art.  104), 
hence  the  hyperbola  has  an  asymptote  passing  through  the 
centre.  Since  the  curve  is  symmetrical  in  reference  to  the 
axis  of  X,  it  will  have  two  asymptotes  passing  through  the 
centre. 

Many  other  curves  have  asymptotes,  but  their  properties 
are  more  conveniently  investigated  by  higher  analysis. 

157.  Equations  to  the  Asymptotes  of  the  Hyper- 
bola.— The  equation  to  the  tangent  line  of  an  hyperbola  is, 
(Art.  102), 

¥x'         I? 

y  =  -T^^ r- 

(^y       y 

But  from  the  equation  of  the  curve  we  have,  (Art.  78,  Eq. 

(«2)), 


[--]U^'[i-^]; 


and  this  value  substituted  in  the  preceding  equation  gives, 
6  1  ab 

y=    ±-X  ■= =:^,    =F       i^ =iL 


r-sj  -['-fl' 


108  come  SECTIONS 

11  x'  =  00,  this  becomes 


[16a 


(1) 


which  is  the  equation  of  the  asymptotes.  The  plus  sign 
belongs  to  the  asymptote  above,  and  the  minus  sign  to  that 
below  the  axis  of  ,/'.  The  asymptotes  are  equally  inclined  to 
the  axes  of  the  curve. 

158.  Equation  of  Condition  for  Asymptotes. — Con- 
struct a  rectangle  on  the  axes,  DD  and  EG 
^^yj//     being  its  diagonals.     Let  DC  A  =  a ;  then 


tana  = 


DA 

CA 


and  tan  ACG  =  —  ->y-.  = =  tam ECA  : 

CA  a 

which,  compared  with  the  coefficient  of  .v  in  equation  (1), 
shows  that : 

The  asymptotes  of  the  hyperbola  coincide  with  the  diagonals 
of  the  rectangle  coiistnicted  on  the  axes. 

Letting  these  diagonals  be  the  asymptotes,  and  ACQ 
=  >^,  and  c  =  V^^  +  If=  CD ;  we  have 


sm  a  = 


sin  y^  = , 

c 


a  a       ^ 

cos  a  =  —  ,       cos  p  =  — 

c  c 


from  which  we  find 


a^  sin^  a  —  V  cos^  «r  =  0 ; 
a^sin^/S-ft^cos^yS^O; 

0?  sin  a  sin  yS—  }?  cos  a  cos  /?  =  — 


a^  +  6?' 


(2) 
(3) 

(4) 


which  are  the  required  equations. 


159,  160.]         THE  HYPERBOLA  AND  ITS  ASYMPTOTES.      109 

159.  Equation  to  the  Hyperbola  referred  to  its 
Asymptotes.— The  equation  to  the  hyperbola  referred  to 
oblique  axes,  the  origin  being  at  the  centre,  is  found  by- 
changing  V^  into  —Ir  in  the  equation  of  Article  127,  and 
hence,  after  dropping  the  accents,  is 

(<2^sin^/5  —  1/  cosV^)  y'+  (a- siii^  a  —  h^ cos,'^  a)  a^  )  _  _  <,p  .^-x 
+  {a-  sin  a  sin  (5  —  h~  cos  a  cos  /i)  ^xy  ) 

Combining  this  equation  with  the  equations  of  condition 
given  in  the  preceding  article,  we  have 

xy  =  l{a'  +  Jr),  (6) 

•which  is  the  required  equation.  It  appears  that  the  rectan- 
gle under  the  coordinates  is  constant.  Let  the  area  of  this 
rectangle  be  represented  by  T<?,  then  we  have 

xy  =  Jc".  (7) 

in  which  x  and  y  may  be  both  positive  or  both  negative. 

For  the  conjugate  hyperbola,  we  have,  (since  ,/'  or  y  becomes 
negative), 

xy=-  k\  (8) 

For  the  equilateral  hyperbola,  a=b,  and  we  have 

xy=±  ha\  (9) 

In  these  equations  x  =  0  for  y=co,  and  y  =  0  for  a:;  =  oo . 

The  asymptotes  are  sometimes  called  self-conjugates  ;  and 
equation  (7)  the  equation  of  the  hyperbola  referred  to  its 
self-conjugates. 

160.  Angle  between  the  Asymptotes  in  terms  of 
the  eccentricity. — We  have  found  for  the  eccentricity,  (Art 
93), 


and,  from  Article  158, 


c 
6=  -, 


a 
cos  a  =  - ; 

c 


.'.sec  a  =  e'. 


no 


CONIC  SECTIONS. 


[161. 


in  whicli  a=DGA.     But  DCG  =  ^DCA',  and  if  DCG  -=  cp, 

we  have 

9>  =  2sec-^e.  (10) 

For  the  equilateral  hyperbola,  this  becomes 

^  =  2sec-V2  =  90°, 

that  is  :  The  asymptotes  of  an  equilateral  hyperbola  are  mutually 
perpendicular. 

161.  Problem. — To  firid  the  area  of  the  parallelogram  con- 
structed on  tJie  coordinates  of  any  point  of  an  hyperbola  referred 
to  its  asymptotes. 

Let  P  be  the  point  whose  coordinates  are  x=  CG,y  = 
GP,  and  let  EGG  =  cp ;  then  will  the  area 
of  the  required  parallelogram  ECGP  be 

GG  sin  q) .  GP  =  xy  sin  cp 

=  i(aH  &')  sin  ^. 

Construct  the  rectangle  AD  PC  on  the 
semi-axes ;  its  area  will  be  ah.     Draw  the 
diagonal  AP ;  it  will  be  parallel  to  CG  and 
UP.    The  area  of  the  triangle  APC  will  be 

^^5.  00  sin  CO  A, 


Tig.  130. 


or 


or 


^.  VoFTW.  I  Va"  +  b^  sin  cp, 
I  (a^  4-  6^)  sin  ^ ; 


which,  compared  with  the  expression  above,  shows  that  it  is 
equal  to  the  area  of  the  parallelogram  GGPE,  but  it  is  also 
equal  to  one-half  the  area  of  the  rectangle  AD  PC',  there- 
fore 

xy  sin  q)  =  \ab ; 

that  is :  The  area  is  constant  and  equal  to  one-eighth  the  red- 
angle  on  the  aoces. 


163, 163.]     THE  HYPERBOLA  AND  ITS  ASYMPTOTES. 


Ill 


162.  Equation  to  any  Chord  referred  to  the  Asymp- 
totes.— Let  x'y'  and  x"y"  be 
the  extremities  of  the  chord 
PP',  then  will  the  equation 
of  the  chord  be,  (Art.  40), 

y-y  _y"-y' 


x  —  x 


But  the  equation  of  the  curve 
gives 

x"  ' 


xy  =x  y  .:y 


which,  substituted  in  the  pre- 
ceding equation,  gives 


Fig.  131. 


xy' 


y-y  =' 


X'  —  x 


~.(x-x'); 


or 


y-y'  =  -^,(^-^')y 


(1) 


which  is  the  required  equation.     It  may  also  be  written 


x  —  x'       y 
x"         y' 


(2) 


163.  Equation  to  the  Tangent  of  the  Hyperbola 
referred  to  its  Asymptotes. — Let  the  chord  pass  through 
P'  continually  as  that  point  moves  along  the  curve  and 
finally  becomes  consecutive  to  P ;  then,  ultimately,  will  x'  = 
x",  and  y'  =  y"  ;  and  equations  (1)  and  (2)  of  the  preceding 
Article  become 


or 


y-y'=  -^{x-x'\ 


x'      y 


(3) 
(4) 


either  of  which  is  the  required  equation.  It  is  also  the  equa- 
tion of  the  tangent  to  the  conjugate  hyperbola. 


112  come  SECTIONS.  [164, 165. 

164.  Intercepts  of  the  tangent  referred  to  Asymp- 
totes.— In  equation  (4)  make  y  =  0,  and  we  have 

or :  The  suhtangent  is  bisected  by  the  ordinate  of  contact. 
If  a?  =  0  in  equation  (4),  we  have 

CB  =  2y'    .'.  CE=EB, 

or :  The  intercept  on  the  axis  of  ordinaies  is  bisected  by  the  ordi' 
note  of  contact. 

Also :  The  portion  of  the  tangent  induded  between  the  asymp- 
totes is  bisected  at  the  point  of  contajd. 

By  comparing  these  results  with  that  found  in  Article  161, 
we  find  that :  Twice  the  area  of  the  parallelogram  constructed 
on  the  coordinates  of  any  point  equals  the  area  of  the  triangle  of 
which  the  tangent  to  the  point  is  one  side,  and  the  intercepts  on  the 
a,symptotes  the  other  two  sides  of  the  triangle. 

165.  Problem. — To  find  the  relation  of  the  segments  NP 
and  N'P'  of  the  secant  NN',  (Fig.  131),  contained  between  the 
a,symptotes  and  the  curve. 

In  equation  (2)  of  Article  162,  if  x  =  0,  we  have,  (observ- 
ing that  x'y'  =  x"y"), 

CN''  =  y  =  y"+y'; 
but  I)'N'  =  CN'-CD'; 

.'.  D'N'  =  y"  +  y--y"  =  y'  =  DP, 

In  a  similar  manner  we  may  prove  that 

DN=D'P'. 

But  the  triangles  Z)PiV and  D'P'N'  are  mutually  equian- 
gular; hence 

PN=P'JV'; 

or  :  The  segments  of  any  chord  contained  between  the  curve  and 
its  asymptotes  are  equal. 


166.] 


THE  HYPERBOLA  AND  ITS  ASYMPTOTES. 


113 


This  principle  furnishes  an  easy  mode  of  constructing  an 
hyperbola  when  the  asymp- 
totes and  one  point  of  the 
curve  are  given.  Let  Tv^ 
and  T-1  be  the  asymp- 
totes, and  a,  any  point  on 
the  curve.  Draw  radial 
lines  a  1,  rt2,  ad,  etc.,  and 
prolong  them  to  Tv^.  Lay 
off  v-iCis  equal  to  ctd,  v^O'i 
=  a%  etc.,  then  will  Og, 
Oa*  etc.,  be  points  on  the 
curve. 

166.  Tangents  at  the  extremities  of  Conjugate  di- 
ameters meet  on  the  asymptotes.  — Let  CA  be  any  diam- 
eter, its  equation,  referred  to  the  axes  x  and  y,  will  be,  (Art. 


Fig.  132 


40,  Eq.  (■/)), 


y'x  -  x'y  =  0, 


(CA) 


in  which  x'  and  y'  are  the  coordinates  of 
A.  The  equation  of  the  tangent  NA  pass- 
ing through  the  same  point  is,  (Art.  163> 
Eq.  (4)), 


y  _ 


-,+  ^,  =  2. 
X      y 


{AN) 


Fig.  133. 


To  find  the  equation  of  the  tangent  BlSf,  it  is  necessary 
to  find  the  coijrdinates  of  B.  The  diameter  CB  being  con- 
jugate to  CA,  will  be  parallel  to  AN,  (Art.  124),  and  hence 
its  equation  will  be  of  the  same  form  as  that  of  {AN),  but 
since  it  passes  through  the  origin,  its  absolute  term  will  be 
zero,  hence  the  equation  of  CB  will  be 


X      y      f. 

x'     y 


{CB) 


which  combined  with  the  equation  of  the  conjugate  hyper- 
bola 

xy  =  —  J{?  =  —  x'y\ 

gives,  by  elimination, 

x  =  CI)  =  Tx',  yr=DB=±y'\  (B) 

which  shows  that  DC  =  —  Ca,  and  Aa  =  BD. 


114  CONIC  SECTIONS.  [167. 

The  equation  of  the  tangent  BN  will  be  of  the  same  form 
as  that  of  NA ;  hence  substituting  in  equation  {AN)  the 
values  of  the  coordinates  of  the  point  B,  as  found  above, 
that  is,  y'  =y'  and  x  =  —  x,  we  have 

--,  +  ^;=2,  {BN) 

X'      y 

which  is  the  required  equation. 

To  find  the  intersection  of  the  tangents  AN  and  BN^ 
combine  equations  {AN)  and  {BN),  and  eliminate  x  and 
y  successively,  and  we  find 

y  =z^y'  =  CN,  and  x=Q. 

This  value  of  x  shows  that  the  intersection  is  on  the  axis 
of  y,  that  is,  it  is  on  the  asymptote. 

This  analysis  also  shows  that : 

The  diagonals  of  all  parallelograms  constructed  on  a  pair'  of 
conjiigate  diameters  coincide  with  tlie  asymptotes  of  the  hyper- 
bola. 

The  value  of  y  shows  that  the  diagonal  CN  of  the  paral- 
lelogram constructed  on  the  semi-diameters,  is  double  the 
ordinate  aA  of  the  extremity  of  the  conjugate  diameter  A' A. 

Polar  Equations  of  the  Conic  Sections. 

167.  General  Equations. — Let  P  be  any  point  on  the 
curve,  F  the  focus,  which  is  also  taken  as  the  pole,  2p  the 
lotus  rectum,  BE  the  directrix,  e  the  eccentricity,  p  =  FP  = 
the  radius  vector,  and  cp  =  PFA,  the  variable  angle  meas- 
ured from  the  nearest  vertex.  Then  according  to  Article 
120, 

FP  =  e.BD  =  e{BF-FD)=e  (^^  -  Fd\ 
But      FB  =  PF  cos  PFD  =  pco&(p, 
therefore,      FP  =  p  =  e(^— pcoscp); 

Fi<^134.  .'.   p=  ^ (1) 

l  +  ecos9>* 


168-170.]  POLAR  EQUATIONS.  115 

which  is  the  polar  equation  of  any  conic  section,  the  pole  being  at 
the  focus  and  the  variable  angle  measured  frorn  tlie  nearest  vertex. 
If  ^  be  measured  from  the  remote  vertex,  cos  q)  in  the 
preceding  equation  becomes  negative,  and  we  have,  (desig- 
nating the  angle  by  tp'), 

P=^—..  (2) 

1  —  e  cos  (p 

If  the  angle  cp  be  measured  from  a  line  which  passes 
through  the  focus  F  and  makes  an  angle  yS  with  the  axis  AF 
of  the  curve,  (/?  being  positive  in  reference 
to  the  line  of  reference),  we  have 

^"^1  +ecos(<7>-/:^)' 

which  is  a  more  general  equation  of  a  conic 

168.  Polar  Equation  to  the  Parabola. —  For  this 
curve,  e  =  1,  and  if  the  pole  be  at  the  focus,  the  initial  line 
coinciding  with  the  axes,  equation  (1)  becomes 

P^,J         ,  (4) 

1  +   cos  q) 

which  is  the  required  equation. 

169.  Polar  Equation  to  the  Ellipse. — Let  the  pole  be 
at  one  of  the  foci,  and  q)  measured  from  the  nearest  vertex. 
We  have  e<  1,  and  2p  =  2a(l-e2),  (Art  95,  Eq.  (2)),  and 
equation  (1)  becomes 

a0^z£L  (5) 

'^       1  +  e  cos  (p  ^ 

which  is  the  required  equation. 

170.  Polar  Equation  to  the  Hyperbola.  —  The  condi- 
tions being  the  same  as  for  the  ellipse,  except  that  e  >  1,  we 
have 

p=<^-^\  (6) 

^      l+ecos<p  ^ 

for  the  required  equation. 


116 


CONIC  SECTIONS. 


[171-173. 


171.  Discussion  of  Equation  (4). — If  (p  =  0,  p  =  Ip, 
whicli  is  the  distance  from  the  focus  to  the  vertex  of  the 
parabola.  ....  If  r/>  =  90^  p  =  p,  which  is  one-half  the 
ordinate  through  the  focus  and  is  one-half  the  latus  rectum, 

as  it  should  be If  </>  =  180"',  p  —  cc,  hence  the  curve 

does  not  cross  the  axis  in  that  direction If  cp=lSO°—i, 

i  being  any  value  however  small,  p  will  have  a  definite 
value ;  hence  if  a  line  be  drawn  from  the  focus  in  the  oppo- 
site direction  from  the  vertex,  making  any  angle  however 
small  with  the  axis,  it  will  meet  the  curve  at  some  point. 
.  .  .  .  If  cp  =  ^lQ\  P=p,  as  it  should;  and  if  qj  =  360% 
p  =  ^p>  which  is  the  same  as  for  (p  =  0°,  as  it  should  be. 

172.  Discussion  of  Equation  (5),  or  p=q — ^^ ^. 

^  '  1  +  e  cos  (p 

If  <^  =0^  p  =  a(l  —  e)=  a  —  ea  =  A'C  — 
FC  =  A'F. Ji  ip=90%  p  =  a{l- 


=  FP For  cp  =1  cos-^ 


FG 


FB 


)= 


ea 


cos  ^—  — ,  we  find  />  =  a,  as  it  should 

If    ^-:180°,   p  =  a{\-¥e)  =  FA If    9?  =  360°,    p  = 

a  (1  —  e)  which  is  the  same  as  for  q)  —  0°. 

173.  Discussion  of  Equation  (6),  or  p=:i— 5^ ^,  for 

^  ^  ^'  1  +  ecos^ 

the  hyperbola.  Since  e>  1,  the  numerator  is  essentially  posi- 
tive, and  the  denominator  will  be  negative  when  cos  qj  is  neg- 
ative and  >  1.     For  qi  —  0%  p  =  a  (e  —  1)  =  F'A For 

q>  =  90°,  p  =  a{^-V)^  FP. 


When  q>=  cos 


■■(-.■) 


the 


Fig.  137. 


radius  vector  becomes  parallel  to 
the  asymptote,  and  the  equation 
gives  p=  cc;  hence  the  positive 
radius  vector  will  not  cut    the 

curve.  As  9?  >  cos~Y  —     )  the  ra- 


■(-;) 


dius    vector    becomes    negative,    and    the    branch    P"  A 


174,  175.]        POLAR  EQUATION  TO  TEE  ELLIPSE.  117 

will  be   described If    <7J  =  180°,    p=  —  a{e  +1) 

=  F'  A,  and  from  this  point  the  branch  AP '"  will  be  de- 
scribed  until    Qi  =  180°  +  cos  "^  -  when   the  radius   vector 

e 

will  be  parallel  to  the  asymptote   CE,  and  will  be  infinite. 
When  it  passes  this  value  it  again  becomes   positive  and 

will  trace  the  branch  FA' Yor  q)  —  270°,  p  =a(e-— 1) 

=F  F' For  (^  =  360°,  p  =  a(e  -  1)  =  F'A. 

174.  Polar  equation  to  the  Ellipse,  the  Pole  being 
at  the  centre. — The  equations  for  transformation  from  rec- 
tangular to  polar  coordinates,  the  pole  being  at  the  origin, 
are,  (Art.  55), 

X  —  pcos  {(p  +  a),  y  =  P sin  {cp  +  a) ; 

and  if  the  initial  line  coincides  with  the  axis  of  x,  these  be- 
come 

x=  p cos  qi,  y=P sin  ^ • 

These  values  substituted  in  the  axial  equation  to  the  ellipse, 
(Eq.  (aO,  Art.  69),  give 

a^  f?  sin-  (p  -\-¥  p^  cos^  (p  =  a^W\ 
ah 
Va"  sin^  (p  +  }?  cos^  cp 

which  is  the  required  equation. 

175.  Polar  Equation  to  the  Hyperbola,  the  Pole 
being  at  the  centre. — Changing  h  to  &  V—  1  in  the  pre- 
ceding equation,  gives 

oib 


V—  <^  sin^  (p  +  W COB^  qi* 
which  is  the  required  equation. 
If  <p  =  0°,  then  p=  ±a; 


cp  =  90°,         p=  ±b  V'^T; 

cp  =  tan ~^- ,  p=  CO  ; 
a 

the  last  value  of  (p  being  the  inclination  of  the  asymptote. 


118  CONIC  SECTIONS.  [175. 


EXAMPLES. 

1.  What  is  the  polar  equation  of  the  parabola  whose  rectangular 

equation  is  y*  =  8  a;  ?    What  is  the  length  of  the  radius  vector  for  ^  =  0% 

4o\  60°,  90%  120"  ? 

4  ,-8 

Am.  p= ;  2;  8-4  /v/S  ;  s  ;  4;  8. 

1  +  cos  qj  o 

2.  Required  the  polar  equation  of  an  ellipse  whose  axes  are  8  and  6 
respectively,  the  pole  being  at  one  of  the  foci. 

3.  What  is  the  polar  equation  of  an  hyperbola  whose  transverse  axis 
is  8,  and  the  distance  between  the  foci  is  12  ?  What  will  be  the  value 
of  piox  cp  =  0°,  90%  120%  180°  ? 

4.  If  a  comet  moves  in  a  parabolic  orbit  having  the  sun  at  the  focus, 
and  is  150,000.000  miles  from  the  sun  when  the  radius  vector  makes  aa 
angle  of  90°  with  the  axis ;  how  near  will  it  approach  the  sun  ? 

Ana.  75,000,000  miles. 


[For  a  shorter  course  omit  to  Chapter  VL] 


CHAPTEK  V. 


GENEBAL    DISCUSSION    OF    THE    EQUATION    OF    THE 
SECOND    DEGKEE. 

176.  Rectangular  Equation  to  a  Conic  Section 
having  any  position  in  a  plane. — Let  P  be  any  point  on 
the  arc  of  a  conic  section  whose 
coordinates  are  OK=x,  and  KP 
=  y;F  the  focus,  and  CB  the  di- 
rectrix. Let  fall  the  perpendicular 
PE  upon  the  directrix,  then,  ac- 
cording to  Article  120,  we  have 
FP  -r-  PE  —  e  =  the  eccentricity. 
It  is  required  to  find  FP  and  PE 
in  terms  of  x  and  y  and  known 
quantities.  Let  the  coordinates 
of  the  focus  be  OH=m,  and  HF 
=  n,  the  distance  OL  of  the  directrix  from  the  origin  be  d, 
and  the  angle  which  the  axis  AF  of  the  curve  makes  with 
the  axis  of  x,  which  equals  LOX,  be  6;  then 

DP:=  OK-  OH=x-m\ 

FD  =  FH-  DH=  n-y; 

.'.  FP^={x- mf  +{y-  n)\  (1) 

Draw  KG  parallel  to  GB,  and  note  the  point  J  where  it 
cuts  PE,  then 

FE=PJ+  OG-  0L  =  y8m  JKP  +  x  cos  GOK-d 
=  y  sin  6  +  a;  cos  B  —  d. 


120  GENERAL  DISCUSSION  OF  THE  [177. 

But,  according  to  Boscovich's  definition  of  a  Conic,  (Art. 
120),  we  liave 

hence  equation  (1)  becomes 

{x  —  mf  +  {y  —  nf  =  e^  {y  sin  6  +  x  cos  6  —  dfi 
expanding  and  reducing  gives 

1-e^cos'^    .      2e^sin^cos^  l-e^sin''^     o 

nr  +  TT  —  rdr  .       m-+n  —  e-cr 

which  is  the  required  equation. 

177.  Every  Equation  of  the  Second  Degree  be- 
tween tTvo  variables  may  represent  a  Conic  Sec- 
tion.— The  general  equation  of  the  second  degree  may  be 
written,  (Art.  92), 

Aa^  +  2Hxy  +  By''  +  "IGx  +  "hFy  +0=0; 

but  since  this  equation  may  be  divided  by  any  one  of  its  co- 
efficients, thereby  making  the  coefficient  of  that  term  unity, 
it  will  be  equally  general  if  we  make  C  unity.  Hence  we 
have  for  the  general  equation, 

Ai?  +  "IHxy  +  By^  +  "hGx  +  "^Fy  +  1  =  0; 

in  which  there  are  five  arbitrary  constants,  independent  of 
each  other.  In  the  equation  of  the  preceding  Article  there 
are  also  five  arbitrary  constants,  viz.,  m,  n,  d,  e,  and  6,  in- 
dependent of  each  other.  If  now  the  coefficients  of  the  cor- 
responding terms  of  the  preceding  equation  and  of  equation 
(2)  of  the  preceding  Article  be  placed  equal  to  each  other, 
we  have 

1-^cos^.  P_    l-e^sin^^ 


rr?  -{-r^  —  e^cP  '  m^  +  n^  —  ^d^  * 

jj_       e^ sin  (9 cos  (9  ^  _e^dcoa  0  —  m 

TTi'  +  n'-e'd''  m^  +  w^-e'd^' 

r,      e^dsind—n 


178,179.]       E2UATI0N  OF   THE  SECOND  DEGREE.  121 

If  any  values  whatever  be  assigned  to  A,  B,  H,  G,  F,  tlie 
values  of  m,  n,  d,  e,  and  B  may  be  found  by  means  of  these  five 
equations,  and  the  latter  quantities  determine  the  character 
and  position  of  a  conic  section  ;  hence,  Every  equation  of  the 
second  degree  represents  some  conic  section. 

178.  General  Test. — Subtracting  the  product  of  A  times 
B,  of  the  preceding  Article,  from  the  square  of  H,  gives 

H^-AB= 

[e*  sin^  d cos'  ^- (1  -e^  cos'  6)  (1  -^  sin^^)] 

1 

(m^  -f-  w^  —  e^  d^f 

=  [e'  sin'  e  cos^  ^  -1  +  (sin^^ + cos^  6)e'-e'  sin^  6  cos^  6] 
1 

e'-l 


(m'  +  n'-e'd^Y' 

the  denominator  of  which  is  essentially  positive,  and  there- 
fore the  sign  of  the  second  member  will  be  the  same  as  that 
of  the  numerator  of  the  fraction.  When  e  >  1  it  will  be  pos- 
itive, and  negative  when  e  <  1.  Hence,  according  to  Article 
93,  we  have 

for  the  ellipse  .  .  e  <  1,  and  H^  —  AB  <  0 ; 
for  the  parabola  .  e  =  1,  and  H^  —  AB  —  0 ; 
for  the  hyperbola  e  >  1,  and  H^  —  AB  >  0. 

The  species  of  the  locus  represented  by  the  general  equa- 
tion of  the  second  degree,  therefore,  depends  upon  the  coeffi- 
cients A,  B,  and  H,  and  is  an  ellipse,  parabola,  or  hyperbola 
according  as  H^  —  AB  is  negative,  zero,  or  positive. 

179.  To  cause  the  term  containing  xy  to  disappear 
from  the  general  equation. — The  coefficient  of  xy  is,  (Art. 
177), 

n^T 2e^sin^cos^ 

^^-~m'  +  n'-e'd'' 


122  GENERAL  DISCUSSION  OF  THE  [180,  181 

which  will  reduce  to  zero  for  G=  0°  or  90°.  The  former 
value  makes  the  axis  of  x  parallel  to  the  axis  of  the  curve, 
and  the  latter,  perpendicular  to  it.  But  changing  the  coordi- 
nate axes  cannot  change  the  character  of  the  locus ;  hence  by 
transforming  the  equation  so  that  the  axis  of  x  will  be  paral- 
lel or  perpendicular  to  the  axis  of  the  curve,  the  general 
equation  becomes 

Ax"  +  Bf  +  2Gx  +  2Fy  +  1  =  0, 

and  since  this  transformation  is  always  possible,  this  equa- 
tion includes  all  varieties  and  species  of  conic  sections. 

Take  the  axis  of  x  parallel  to  the  axis 
of  the  curve  ;  then  B  —  0,  and  the  values 
of  the  coefficients,  (Art.  177),  become 

l_e2  1 

-4  ^  — o~:    ~o ms  »      x>  = 


Fio.  139. 


m^  +  n^  —  e-(^ '  m^ -{- n^  — 

H=0; 


^  ^d  —  m  r,  —  w 

G=    ,  .    , — :^;    F  = 


180.  To  cause  the  coefficient  oty  to  disappear. — This 
condition  requires  that  F  be  zero  in  Article  177, 

.♦.  ^  =  0,  and  n  =  0, 

which  makes  the  axis  of  x  coincide  with  the  axis  of  the 
curve.  This  condition  also  reduces  H  to  zero,  hence  the 
general  equation  becomes 

Ja^  +  By^  +  2Gx  + 1=0; 

which  involves  dU  varieties  of  conic  sections.  The  values  of 
the  coefficients  become 

J         1  —  e^  „  1  ^       e^d  —  m 

Jj  =  — 5 5-T5  ;       ir  = 


rri^-^dj"'  rri'-^d?'  m^-e^^' 

181.  Remark. — When  JJand  i^are  zero,  no  other  coejffl- 
dents  can  generally  be  zero.    The  value  of  A  cannot  also,  gen- 


183.]  EQUATION  OF  THE  SECOND  DEGREE.  123 

erally,  be  zero,  for  if  it  is,  e  must  be  "unity,  wliicli  is  true  only 
for  the  parabola ;  B  cannot  be  zero  unless  m  or  d  is  infinite, 
neither  of  which  would  represent  a  finite  curve  ;  and  G  cannot 
generally  be  zero,  for  if  it  were  we  would  have  c?  =  w  in  the 
parabola,  which  would  cause  the  directrix  to  pass  through 
the  focus,  a  condition  which  is  not  true  of  the  common 
parabola. 

Varieties  of  Conic  Sections. 

182.  Varieties  of  the  Ellipse. — For  this  case  we  have 
to  determine  all  the/orms  of  loci  represented  by  the  equation 
Ax^  +  Bi/  +  2Gx  +  l  =  0,  when  ^^  _^b  <  0. 

Since  H  is  zero  in  this  equation,  we  have  —  AB  <  0,  hence 
A  and  B  must  be  finite  and  have  the  same  signs.  The  ori- 
gin may  be  so  taken  that  m  =  e^d,  in  which  case  G  =  0,  and 
the  equation  becomes 

Ax'  +  By' +  1  =  0,  (1) 

•which  involves  all  the  varieties  of  the  ellipse. 

If  A  and  B  are  essentially  negative,  this  equation  be- 
comes 

Ax'  +  By'=+  1,  (2) 

which  is  the  equation  of  the  ellipse  referred  to  its  axes, 
(Art.  69).     If  A  and  B  are  essentially  positive,  we  have 

.=  |/^i^,  (3) 

which  is  imaginary  and  the  equation  represents  no  real 
locus,  but  we  may  say  that  it  represents  an  imaginary  lo- 
cus. If  ^  =^  and  both  are  negative,  we  have  Ax^  +Ay^  =1, 
which  is  the  equation  of  the  circle,  (Art.  57).  Substituting 
m  =  e^d  in  the  values  of  A  and  B  of  Article  180,  we  have 

'^-'~'7d^(l^^)-~  e'd^'  e'd'il-^)     ^^ 

Substituting  these  values  in  equation  (1),  multiplying 
through  by  (1  —  e"),  and  representing  the  coefficients  of  a^ 
and  y'  respectively  by  A'  and  B',  gives 

A'x'  +  By  +  {l-e')=0, 


124  GENERAL  DISCUSSION  OF  TEE  [183. 

If  now  the  absolute  term  be  zero,  or  e^  =  1,  the  equation 
reduces  to  the  form 

Aq?  +  B'f  =  0. 

But  an  examination  of  equation  (4)  shows  that 

1  — e^ 

which  is  zero  when  (?  =  1,  hence  the  equation  becomes 
O.x'  +  By^O, 

which  is  satisfied  for  ±  x  indetermiTiate  and  ?/  =  0 ;  and  hence 
is  the  equation  of  the  axis  of  x ;  hence  a  straight  line  is  a 
particular  case  of  an  ellipse.  But  when  e  =  +  1,  the  locus  is 
a  parabola,  (Art.  93) ;  hence  the  ellipse  and  parabola  ap- 
proach the  same  right  line  as  one  of  the  limits  of  those  curves. 
There  are,  therefore,  five  varieties  of  loci  embraced  in  the 
general  equation  of  the  second  degree,  -^hich  fulfil  the  con- 
dition H^—  AB  <  0,  and  hence  are  called  varieties  of  the 
ellipse  ;  viz.,  the  Ellipse  proper,  the  Circle,  the  Point,  the 
Right  Line,  and  an  Imaginary  Locus. 

183.  Varieties  of  the  Hyperbola.  — These  are  in- 
cluded in  the  equation  Ax'  +  Bif  +  2  (ro?  + 1  =  0,  (Art.  180), 
under  the  condition H'^—  AB  >  0.  Since  H  —Om  this  equa- 
tion, A  and  B  must  have  contrary  signs  so  that  their  pro- 
duct shall  always  be  negative,  and  hence  —  AB  be  always 
positive.  Taking  the  origin  so  that  m  =  e^d;  in  which  case 
G=0,  the  equation  becomes 

Ax'-By^-l^O; 

in  which  A  and  B  must  have  the  same  sign,  since  the  sign  of 
one  term  is  changed.  If  A  and  B  are  essentially  negative, 
this  equation  becomes 

Ax"  -By^=  +  1,  (1) 

which  is  the  equation  to  the  ordinary  hyperbola  (or  x-hyper- 


18-1.]  EQUATION  OF  THE  SECOND  DEGREE.  125 

hola),  (Art.  78).  If  A  and  B  are  essentially  positive,  this 
equation  becomes 

■which  is  the  equation  to  the  conjugate  hyperbola  (or  y-hyper- 
bola).     Jl  A  —  B,  we  have 

Ax-  —  Aif  =  ±  1 ; 

which  is  the  equation  to  the  equilateral  hyperbola.  Substi- 
tuting the  value  of  m  —  e^d  in  the  values  of  A  and  B,  (Art. 
180),  they  become  the  same  as  for  the  ellipse.  Substi- 
tuting their  values  in  equation  (1),  multiplying  through  by 
e^cP  (1  —  e^),  and  representing  the  coefficients  by  A'  and  B' 
(observing  that  they  have  contrary  signs),  and  making  d  =  0, 
e  and  A'  will  be  indeterminate,  and  we  have 


^V-J5y  =  0;     .•.x=±\/ 


B 


which  is  the  equation  of  two  right  lines  passing  through  the 
origin.  If  d  be  zero  or  finite,  and  e  =  ±  1,  in  which  the 
upper  sign  is  characteristic  of  the  parabola,  the  value  of  A' 
for  both  values  of  e  becomes  zero,  and  the  equation  be- 
comes 

0.x'-By  =  0; 

which  is  satisfied  for  +  x  indeterminate  and  y  =  0,  and  hence 
is  the  equation  to  the  axis  of  x.  This  same  right  line  is, 
therefore,  a  limit  both  of  the  hyjjerbola  and  parabola.  Hence, 
the  condition  H'  —  AB  >  0,  in  the  general  equation  of  the 
second  degree,  gives  four  varieties  of  loci,  and  are  called 
varieties  of  the  hyperbola ;  viz.,  the  Common  Hyperbola,  the 
Equilateral  Hyperbola,  Two  Bight  lAnes  intersecting  each  other, 
and  One  Right  Line. 

184.  Varieties  of  the  Parabola. — Eesuming  equation 
Ax''  +  By''  +  '2,Gx  +  l  =  Q, 
and  the  analytical  condition  for  the  parabola,  which  is 
H'-AB  =  0; 


126  GENERAL  DISCUSSION  OF  TEE  [184. 

we  observe  that,  since  H  is  zero  in  tlie  former  equation, 
either  A  or  B,  or  both  A  and  B,  must  be  zero. 
If  ^  =  0,  we  have 

By-  +  '2.Gx  +  l  =  Q. 

But,  for  the  parabola,  e  —  1,  and  the  equations  in  Article 
180  become 

j-n.      p-      ^        .     r-    ^~'^    - ?— 

^~"'     ^-  m^-d^'     ^-  ra^-d-^-      m  +  d' 

SiA)stituting  these  values  in  the  preceding  equation  and 
multiplying  by  m^  —  d\  gives 

y"  +  2(c?  -  m)x  +  m^-d^=Q. 

The  absolute  term  is  zero  when 

m^  =  rf^ ;  or  m  =  ±d. 

For  m  =  4-  (f,  the  preceding  equation  becomes 

which  is  the  equation  of  the  axis  of  x.  But  when  m=  ■{■  d 
the  directrix  passes  through  the  focus,  (Arts.  176  and  179), 
and  this  condition  reduces  the  parabola  to  a  straight  line 
passing  through  the  focus. 

li  m=  —  d,  the  absolute  term  vanishes,  and  the  coeffi- 
cient of  X  becomes  4c?,  and  the  equation  becomes 

y"^  +  4dx  =  0 ;  or  y^  =  —  4dx. 

If  d  be  positive,  y  will  be  real  for  negative  values  of  a;, 
and  imaginary  for  positive  values. 
If  d  be  negative,  it  may  be  written 

y^  =  4dXf 

which  is  the  equation  of  the  common  parabola. 

The  condition  that  m  —  —  d,  places  the  directrix  as  far 
from  the  origin  in  one  direction  as  the  focus  is  in  the  oppo- 
site direction,  which  agrees  with  Article  85. 


184]  EQUATION  OF  THE  SECOND  DEGREE.  127 

If  jB  =  0,  we  have  the /orm 


G      JG^      1 


and  y  indeterminate  ;  but  to  make  B  =  0,  m  or  d  must  be 
infinite,  either  of  which  reduces  A  and  G  to  zero,  hence 
the  preceding  value  of  x  becomes  a?  =  oo ,  hence  the  line  is 
parallel  to  y,  but  infinitely  distant. 

Substituting  e  =  1,  in  the  equations  of  Article  180,  we 

find   that  if  5  =  co  we   have  w?  =  d^ ;   .:  A  =  ^;  and  also 

G  =  ^',  that  is,  both  are  indeterminate. 
If  ^  =  0  and  5  =  0,  we  have 

2(?a^  + 0.2/^  +  1=0, 

which  is  the  equation  of  a  right  line  parallel  to  the  axis  of  y, 

and  at  a  distance  from  it  equal  to  —  ^^-^ . 

AG 

In  the  general  equation 

Ax"  +  2Hxy  +  By-  +  'AGx  +  "AFy  +  1  =  0, 

a  A  =  B=  ±  H,  we  have  H^  —  AB  =  0  (which  character- 
izes the  parabola),  and  the  equation  becomes 

Aix+yf  +  'AGx-^  2Fy  +  1  =  0. 

Making  A  =  B  =  ±  ^in  the  equations  of  Article  177,  and 
at  the  same  time  making  e  =  1,  we  find  that  sin  6  =  cos  6 ; 
.'.6=  45°  ;  hence  the  axis  of  the  curve  cuts  both  coordinate 
axes  at  an  angle  of  45°.  Transforming  the  origin  of  coordi- 
nates to  a  point  such  that  F=  0,  and  G  =  0,  we  find  from 
the  last  equations  of  Article  177  that 

wi  =  n,  d  =  V  2  n,  and  ^  =  oo. 


128  GENERAL  DISCUSSION  OF  TEE 

Ji  F=  (r  we  have 

A{x  +yy+2G(x  +  y)=  -1; 


[185. 


•••  y 


—  X ^  ± 

A 


^  A'      A' 


which  is  the  equation  of  two  real  straight  lines  if  ^  <  G^] 
of  two  imaginary  lines  if  ^  >  G*,  and  of  one  real  line  if 
A=  cc. 

There  are,  therefore,  five  varieties  of  the  parabola,  viz., 
the  Common  Parabola,  the  Imaginary  Parabola,  the  Bight 
Linef  two  ParaUd  Bight  Lines,  and  two  Imxxginary  Bight 
Lilies. 

185.  Illustration. — It  will  be  shown  hereafter,  (Art. 
252),  that  the  intersection  of  a  plane  with  a  cone  will  be 
a  parabola  when  the  plane  is  parallel  to 
an  element  of  the  cone,  as  LB ;  an  ellipse 
if  the  plane  cuts  all  the  elements  of  the 
cone,  as  AE;  an  hyperbola  if  it  cuts  both 
nappes  of  the  cone,  as  DA  '  AH.  Taking 
these  for  granted,  we  may  easily  illus- 
trate all  the  varieties  of  the  conic  sec- 
tions. 

If  the  plane  of  the  ellipse  be  turned 
until  it  is  perpendicular  to  the  axis  of  the 
cone,  the  line  of  intersection  will  con- 
stantly be  an  ellipse,  but  when  the  plane 
becomes  perpendicular  to  the  axis  the  intersection  will  be  a 
circle.  If  the  plane  pass  through  the  apex  it  will  cut  all  the 
elements,  for  they  all  pass  through  that  point,  and  the  inter- 
section becomes  a  point.  If  it  be  turned  until  it  is  tangent 
to  the  cone  along  AB,  the  intersection  becomes  a  straight 
line,  which  is  one  of  the  special  cases  of  an  ellipse.  If  the 
apex  of  the  cone  be  infinitely  remote,  the  cone  becomes  a 
cylinder,  and  the  intersection  AE  is  still  an  ellipse. 

If  the  plane  DA  AH  pass  through  the  vertex,  it  will 
intersect  both  nappes,  but  the  intersection  will  be  two 
straight  lines  intersecting  at  the  apex.    If  the  plane  be  turned 


186.]  EQUATION  OF  THE  SECOND  DEGREE.  129 

about  an  axis  through  the  vertex,  the  two  lines  approach 
the  single  line  VB  as  a  limit,  and  this  is  a  particular  case  of 
the  hyperbola. 

If  the  plane  LR  passes  through  the  apex  and  parallel  to 
an  element  the  intersection  Avill  be  a  rigid  line,  AB.  If  the 
apex  of  the  cone  be  infinitely  distant,  the  surface  becomes  a 
cylinder,  and  the  intersection  of  a  plane  parallel  to  an 
element  will  be  two  rigid  lines  unless  the  plane  be  tan- 
gent to  the  surface,  in  which  case  it  will  be  limited  to  one 
right  line.  If  the  plane  be  parallel  to  an  element,  but  does 
not  cut  the  surface  of  the  cylinder,  the  intersection  will  be 
imaginary,  and  will  represent  the  two  imaginary  right  lines. 

186.  Problem. — Pass  a  conic  through  any  jive  points  in  a  plane,  and 
show  that  only  one  such  conic  can  he  passed. 

Let  (a;,,  y,),  {Xi,  y^),  {x^,  y.,),  (x^,  y^),  and  (a;.,,  y^)  ^^  tte  five  points. 
Substituting  these  values  successively  in  the  general  equation  of  the  second 
degree,  (Art.  177),  we  have  the  five  equations  of  condition 

Axi"  +2Rx,y,  +  By,^  +  20xi  +2Fyi  +1=0 
AX2'  +  2Hx2y2  +By2^  +  2Gx.,  +  2Fy2  +  1  =  0 
Ax:,^  +  2Hx,y.,  +By,'  +  20x^  +  2Fy,  +  1  =  0 
Ax  J  +  2HXiyi  +By^^  +  2GXi  +  2Fy^  +1  =  0 
Ax,-"  +  2nx,y^  +By,^  +  2Gx,  +  2Fy,  +  1  =  0 

from  which  the  value  of  the  five  arbitrary  constants  may  be  found,  which 
values  substituted  in  the  general  equation  Ax^  +  2Hxy  +  By'^  +  Gx  +  Fy 
+  1  =  0  will  give  the  required  equation.  Since  the  unknown  quantities  A, 
H,  etc.,  are  of  the  first  degree,  only  one  set  of  values  can  be  found,  and 
since  the  five  points  give  five  conditions,  they  can  generally  be  definitely 
determined. 


EXAMPLE. 
1.  Determine   the   equation  of   a  conic   section   which   shall  pass 
through    the    points    (2,    1),     (  3  >  2  j  ,      (  ~  3 '  ~  3  )  '      (  ~  3 '  ~  ^)  ' 
find  its  species,  its  eccentricity,  the  position  of  its  directrix, 


V3'      3^  ' 


and  locate  the  curve  in  referenfce  to  the  coordinate  axes. 
9 


130  GENERAL  DISCUSSION  OF  TEE  [186. 

The  equations  of  condition  will  be 

25         20  10 

-g-^  +  yS  +  45  +  y(?  +  4i^^+ 1  =  0  ; 

4         4         14         3 


-^A+4:H+^B-~G-%F+  1=0; 


25^      20^^     4„      10^     4^      ^      ^ 
_^__^+-5+-^-^i^+l=0. 


Solving  these  equations,  give 


and  the  equation  of  the  locus  will  be 

2  14        2 

-x^  +  -^<cy—^y^  +  ga!--2/  +  l  =  0; 

in  which  the  absolute  term  must  remain  positive  since  it  is  positive  in 
Ai-ticle  177. 

To  determine  the  species  of  this  locus,  we  have 

fi-* -^5  =  - --  =  --,  which  is  <0, 

hence  the  locus  is  an  ellipse.     We  now  have,  according  to  Article  177, 

(2) 


i-.'cos*e       _ 

m«  +  %2  -  e*  d^  -      ^'       ^^^ 

1  -  e«  sin*  0 

1 
~3' 

c*  sin  6  cos  0              1 

e^  d  cos  0  —  m 

2 

m«  +  ri«  -  e*  <Z^  -      3'      ^^ 

m^  +n''  -  e^  d^  ~ 

3' 

e^  cf  sin  9  — 

n             1 

7?l«  +  72,2   _  e« 

d^-~S- 

From  (1)  and  (2)  we  have 

l-e«(l-sin*0) 

=:3-3e2sin«e  ; 

.'    ain«  e  : 

2  +e« 

(4) 
(5) 


(6) 


186.]  EQUATION  OF  THE  SECOND  DEGREE.  131 

From  (2)  and  (3)  we  have 

e-  sin  (3  cos  0  =  1  —  6*  sin*  9. 
Squaring,  substituting  sin-  0  from  (6),  we  find 

e^  =  3(1^  -  1) ;  (7) 

.-.  e  =  0.910+.  (8) 

This  value  of  e  in  (6)  gives 


sin  e  =  W'2  +  V2  =  0. 9338  +  ;  .  •.  6  =  67°  30'.  (9) 

Equation  (3)  now  gives 

=  -K3-v^);  (10) 

and  (4)  and  (10)  give 

e^  (Z  cos  6  -  m  =  -  (3  —  VS) ;  (11) 

and  (5)  and  (10)  give 

c«  <Z  sin  0  -  71  =  i  (3  -  VS).  (13) 

Let  ^  =  K3  -  '/2)  =  i  '/S  (  1^3  -  1) ; 

then,  (Eq.  (7)),  e^  =  3  V34 ; 

and  from  Eq.  (9)  cos  9  =  ^  V2A ; 

and,  therefore,  sin  0  =  ^  V^  V%  —  A\ 

and  equations  (10),  (11),  (13),  become 

m«  +  n« -3|/3^(Z^  =  -3^;  (13) 

2A^d  -m=  -2A;  (14) 

2Ad  V2-A-n  =  A.  (15) 

Eliminating  d  and  n  from  these  equations,  we  find 

iA-iV2)m^ 
+  (2^=«  -  4^*  -  A^  V2A  -A^  +  V2A)m 

=  Ia*-  —  A^-2A'V2A-A'+  V2A'. 
2  2 


132 


GENERAL    DISCUSSION  OF  THE 


[187. 


Introducing  the  value  of  A  and  reducing,  we  find 
OT*  —  TO  =  0.68  +,  nearly; 
.-.  m=  1.46+  or-0.46+. 
These  values  in  (14)  give 

<Z  =  2.76or  -3.31; 

and  these  in  (15)  give 

71=  1.81  or  -2.82. 

Making   05"=  m  =1-46,  and  HF=n  =  1.8  +,  locates  the  focus  F, 

and  0^'=:n  =  +2.8  +  , 
^L  E' F'  =  m=  -  0.  46 

locates  the  focus  F'\ 
and  the  line  J.  J.  drawn 
through  the  foci  lo- 
cates the  major  axis. 
It  should  make  an  an- 
gle i^G^X=  67°  30'. 

The  distance  OL 
—d  —  2.7  + "parallel  to 
AA\  gives  the  dis- 
tance of  the  directrix 
LB  from  tlie  origin, 
and  the  line  LB, 
drawn  through  Z,  per- 
pendicular to  AA ' 
gives  one  directrix. 
Similarly,  OL '  —  dz= 
—  3.3  +,  determines 
the  other  directrix. 
The  centre  G  is  mid- 
way between  F  and 
F\  and 

CFAF 

Fig.  141.  AG' AB'^ -^•^^^' 

\S'^ .  Another  Mode  of  Discussion. — The  process  of  deter- 
mining the  position  and  dimensions  of  a  curve  by  the  pre- 
ceding process  is  often  very  lengthy,  so  that  instead  of  re- 
sorting to  it,  the  curve,  in  practice,  is  simply  constructed  by 
points.  Thus  in  the  preceding  example,  in  which  the  equa- 
tion is 

^2         1   2     4        2        -      ^ 


187a.]  EQUATION  OF  THE  SECOND  DEGREE.  133 

we  assume  values  for  y  and  find  x.     We  thus  find 

if     y  =  0,  a^  =  1.86+  =  OX,  or -0.53  + =0X'; 

2/  =  1  =  Oa,  a:  =  2  =  o?>,  or  0  ; 

2/  =  2  =  Oc,  x=  1.66  +  =  cA,  or  +  1.00  =  cd; 

S/=-l  =  0/;  a;  =  1.53 +=/;?,  or -0.86+  =/^; 
etc.;  etc.; 

througli  wliich  points  the  curve  may  be  traced. 

A  general  discussion  may  be  made  without  introducing 
the  eccentricity.     Resuming  the  general  equation 

Aa? +  '2>Hxy  +  By- +2.Gx  +  2Fy +  1  =  0,         (1) 

and  solving  for  y,  we  have 

y=-  —^ ±  i  V (^^  -  AB)j?+  2  {HF-  BG)x+F^-  By 
B  B 

which  generally  gives  two  values  for  y,  one  of  which  exceeds 

Hx  +  F 

the  ordinate  given  by  the  rational  part, = —  ,  as  much 

B 

as  the  other  is  less  than  that  part.     If  the  rational  part  be 

represented  by  yi,  then 

Hx    +    F  fcys. 

is  the  equation  of  a  diameter.     This  diameter  will  be  conju- 
gate to  the  diameter  parallel  to  the  axis  of  y. 

187a.  To  remove  the  coefficients  F  and  G  from  the 
general  equation. — Transform  the  axes  to  a  new  origin,  the 
axes  remaining  parallel.     For  this  case  we  have,  (Art.  49), 

x  =  a^  +  m,  y  =  yi  +  n,  (3) 

(since  m  and  n  may  be  either  positive  or  negative),  and  sub- 
stituting these  values  in  equation  (1)  we  find 

Ax{^  +  2Hxiyi  +  Byi^ + 2{Am  +  En+  G)xi  +  2{Bn+ Hm 
+F)y,  +  G'=0,  (4) 


134  GENERAL  DISCUSSION  OF  TEE  [l^lh. 

in  which  C '  is  the  sum  of  the  absolute  terms.  Making  the 
coefficients  of  x^  and  y^  equal  to  zero,  we  find 

BG-HF  AF-HG  ,.-, 

from  which  the  values  of  m  and  n  may  be  found,  except  when 
H^  —  AB  —  0,  in  which  case  m  and  n  are  either  infinite,  or 
indeterminate.  The  finite  values  of  m  and  n  from  equations 
(5),  substituted  in  equation  (4),  give,  after  dropping  the  sub- 
scripts, 

Ax'-  +  2Hxy  +  By-+  C  =  0;  (6) 

hence,  if  the  coefficients  of  the  first  powers  of  both  x  and  y 
in  the  general  equation  can  be  removed  at  the  same  time,  it 
can  be  done  by  changing  the  origin  to  a  point  whose  coordi- 
nates are  given  by  equations  (5). 

If,  in  equation  (6),  we  substitute  —  x  for  x,  and  —  y  for  ?/, 
it  will  remain  of  the  same  form  and  value  ;  hence  the  locus 
will  have  a  centre  and  the  origin  of  coordinates  will  be  at  that 
centre.  Hence,  also,  for  every  locus  of  the  second  order 
which  has  a  centre,  the  origin  of  coi'jrdinates  may  be  taken 
at  such  a  point  that  the  terms  containing  the  first  powers 
of  the  variable  shall  disappear. 

Comparing  the  value  of  H^  —  AB  with  Article  178,  we 
see  that  the  parabola  has  no  centre,  but  that  the  ellipse  and 
hyperbola  have  each  a  centre. 

187 J.  To  remove  the  term  containing  a;  ?/ from,  the 
General  Equation. — Turn  the  new  axes  through  an  angle 
a.    For  this  we  have,  (Art.  54), 

x  =  x'  cos  oc  —  y'  sin  o', 

y  —  x  sin  oc  ■\-  y'  cos  oi ; 

and  these  values  substituted  in  equation  (6)  give 

{A  QO'^a  +  B  sin'ci'  +  2JT  sin  ix  cos  a)x  ^  \ 

+  {A  sin^«'  +  B  co'i-tx  -^H  sin  a  cos  oc)y'  ^  r  •  (7) 

+  \^{B—A)  sinacosar  +2ZZ'(cosV— sinV)]a;y  +  C"  =  0 ) 

Making  the  coefficient  of  x'y'  equal  to  zero,  we  find 


1876.]  EQUATION  OF  THE  SECOND  DEGREE.  I35 

2  sin  a  cos  ot       ,      ^  %H  .o\ 

2 ^^r~  =  tan  2a  =  — (8) 

and  equation  (7)  will  reduce  to  the  form 

A'x^+  B'y'^+  C'  =  0; 

in  which  A  '  represents  the  coefficient  oi  x'^ ;  and  B\  oiy'^; 
or,  dropping  the  accents,  it  becomes 

Ax'-i-  By-+  (7=0.  (9) 

Hence  the  term  containing  xy  in  the  general  equation  of 
the  second  degree  may  be  made  to  disappear  by  turning  the 
rectangular  axes  through  an  angle  a,  the  value  of  which  is 
given  by  equation  (8). 

After  removing  the  term  containing  xy,  the  general  equa- 
tion becomes 

Aa?  +  By^  +  2Gx  +  2Fy+  C=-  0,  (10) 

and  when  the  locus  has  not  a  centre  we  have  H^  —  AB  =0; 
and  since  His  zero  in  the  preceding  equation,  we  must  have 
either  Aox  B  equal  to  zero,  and  the  equation  becomes 

Ax'  +  2Gx  +  2Fy-\-  C=0) 
or  By'  +  2Gx  +  2Fy  +  C^0.\  ^^^■^ 

The  origin  of  coordinates  may  now  be  changed  to  such  a 
point  as  to  reduce  these  equations  to  the  form 

A,x^  +  2F,y^0,) 
or  Biy'  +  2GtX  =  0.)  ^^"^^ 

which  are  the  well-known  equations  of  the  parabola. 


EXAMPLES. 
1.  Required  the  equation  of  the  conic  passing  through  the  five  points 
(0,  0),  (3,  3),  (8,-6),  (18,-9),  (33,  12),  and  find  its  species. 

9 

Am.  y*  =  -X. 


136 


GENERAL  DISCUSSION. 


[187&. 


2.  Find  the  equation  of  a  conic  passing  through  (—2,  —4),  (1,  2), 
(0,  0),  (3,  -  6),  and  (-  1,  2). 

Am.  1/  =  ±  2x. 


3.  y*  —  2x1/  +  2x'  —  2a;  =  0,  is  the  locus  of 
■what  ?  Does  it  pass  through  the  origin  ?  Does  it 
cut  both  coordinate  axes  ? 


Fie.  14S. 


EXAMPLES.  137 


ADDITIONAL  EXAMPLES. 

Of  Points  and  Lines. 

1.  Find  the  coordinates  of  the  point  which  bisects  the  line  joining 
the  points  whose  coordinates  are  (2a,  —  J),  (— «,  2h). 

2.  Show  that  the  area  of  the  triangle  inclosed  by  the  lines 

X  +  2y— 5  =  0,         2x  +  y  —  7  =  0,        y_a-_i_i  —  0, 
.    3 

3.  Show  that  if  the  whole  area  included  between  the  lines  x  +  y  =  c, 
and  hx+  ay  =ab  and  the  coordinate  axes,  be  bisected  bj-  the  line  which 
joins  the  origin  with  their  point  of  intersection,  then  c  is  a  geometric 
mean  between  a  and  b. 

4.  Prove  that  the  equations  to  the  lines  bisecting  the  angles  between 
the  lines  whose  equations  are 

12a;  +  5y  =  8,  and  dx  —  4y=  3, 
will  be 

99a;  -  27y  =  79,  and  21a;  +  77y  =  1. 

5.  Knd  the  condition  of  perpendicularity  of  the  straight  lines  repre- 
sented by  the  equations 

x  +  {a  +  l)y  +  c=0, 

a(x  +  ay)  +  h  (x  —  h/)  -h  d  =  0. 

6.  Find  the  equation  to  the  line  passing  through  the  origin  and  per- 
pendicular to  the  line  x  +  y  =  2. 

7.  Find  the  perpendicular  distance  of  the  point  (1,-2)  from  the 
line  x  +  y  —  3  =  0. 

8.  Find  the  equation  to  the  line  which  passes  through  the  point 
(a,  J),  and  through  the  intersection  of  the  lines 

*       y      .  a;       y 

ah        '  6       a 

X         y       1      1 
Arts.   —7  —  Ti  = r  • 


138  EXAMPLES. 

9.  Given  the  coordinates  of  the  vertices  of  a  triangle,  to  find  the 
equation  to  the  line  which  joins  the  middle  points  of  two  sides. 

10.  Find  the  tangent  of  the  angle  between  the  lines 

y  —  mx  —  0,  and  my  +  «  =  0, 

when  referred  to  oblique  axes. 

m*  +  1 

An».   —T. — r  tan  <», 

m^—1 

11.  Show  that  the  tangent  of  the  angle  between  the  two  lines  con- 

vex*-4^  C) 
tained  in  the  equation  Ay^  +  Bxy  +  Cx^  =  0,  is .       ^ — . 

12.  Show  that  4j!^  —  12xy  +  9y^  —  4:X+  6y  —  12  ~0  represents  two 
parallel  riglit  lines. 

13.  Show  that  9x*  —  12xy  +  iy^  —  2x  +y  —  3  =  0  does  not  repre- 
sent right  lines,  and  find  what  value  must  be  assigned  to  the  coefficient 
of  X  in  order  that  it  may. 

14.  Find  the  equation  to  the  line  passing  through  the  intersection  of 
X  cos  a  +  y  sin  a  =.  j/,  and  x  cos  ^  +  y  sin /J  =  p'l  and  cutting  at  right  an- 
gles the  line  x  cos  y  +  y  sin  y  =  p". 

15.  The  lines  drawn  from  the  angles  of  a  triangle  to  the  middle  of 
the  opposite  sides  meet  in  one  point.  (For  a  solution  by  Quaternions,  see 
Articles  335-7.) 

16.  Show  that  the  lines  which  bisect  the  angles  of  a  triangle  inter- 
sect in  one  point.     (See  Articles  341-3.) 

17.  Show  that  the  altitudes  of  a  triangle  meet  in  a  point. 

18.  Substitute  numerical  coefficients  in  the  equation  Ax^  +  2Hxy 
+  By^  +  2Gx  +  2Fy  +  C  =  0,  so  that  the  result  shall  represent  two  right 
lines. 

19.  Find  the  polar  equation  to  the  line  passing  through  the  points 
(3,  30°),  and  (2,  60^). 

20.  Construct  the  line  r  cos  {cp—  it)  -\-r  sin  {qt—  \n)  =  2. 

21.  Find  the  polar  equation  to  a  line  jDcrpendicular  to  the  line^  z=  r 
cos  (6  -  60°). 

22.  Construct  the  line  5  =  3rcos  6  —  6r  sin  S. 

23.  Show  that  the  angle  between  the  lines  a  =  4r  cos  6  +  dr  sin  6, 
and  &  =  3r  cos  6  —  4?*  sin  S  is  right. 

24.  Find  the  polar  equation  to  a  line  passing  through  the  point 
(6,  45°)  and  making  an  angle  of  ^7t  with  tlie  initial  line. 

25.  Find  the  point  of  intersection  of  the  lines  p  =  r  cos  (9  —  f  tt),  and 
8r  cos  6  -i-  4r  sin  0  -1-  5  =  0. 

Of  the  Circle. 

26.  Find  the  equations  to  the  straight  lines  joining  the  centre  of  the 
circle  «*+  y^=a',  with  the  points  in  which  the  line  2(x+  y)=  a  meets  it. 

Am.  y  =  \{±  4/7  —  4)3;. 


EXAMPLES,  139 

27.  Determine  the  centre  of  the  circle  x^  +  y^  +  ix  +  Ap  —  1  =  0. 

28.  Find  tlie  equations  of  the  two  circles  which  touch  the  straight 
lines  y±  X  —  0,  and  pass  through  the  i^oint  {h,  F). 

A71S.  x^  +  y^-  -  2y  (2Z;  ±   V'2yl'=^- 2/i~  )=  ^(2^  ±  V'Zk''  -  2h-^  y. 

29.  Show  that  the  locus  of  a  point  whose  distance  from  a  fixed  line 
is  equal  to  its  distance  from  a  fixed  circle,  is  a  parabola. 

30.  Find  the  points  of  intersection  of  the  circle  x'^  +  y^  =  12,  and 
the  line  8.c  -  2y  =  6. 

31.  Find  the  equation  to  the  tangent  of  the  circle  x^  +  y^  —  2y  — 
Bx  =  0,  passing  through  the  origin. 

Arts.   2y  +  iix  =  0. 

32.  Find  the  equation  to  the  circle  which  passes  through  the  origin 
and  the  intercepts  «  =  4  and  y  =  5. 

33.  Show  that  the  portion  of  the  line  x  +  y  =  3a,  which  is  between 
the  coordinate  axes,  is  trisected  at  the  points  where  it  cuts  the  circle 
x^  +  y^  +  a^  =  2a{x  +  y). 

34.  A  CB  =  2R  is  the  diameter  of  a  circle,  CP,  CQ  are  perpendicular 
radii ;  show  that  the  locus  of  the  intersection  of  AP  and  BQ  is  a  circle 
whose  centre  is  iu  the  given  circle,  and  radius  is  V2B. 

35.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances 
from  the  four  sides  of  a  square  is  constant;  show  that  the  locus  of  the 
point  is  a  circle. 

36.  Show  that  the  locus  is  a  circle  when  the  point  moves  so  that  the 
sum  of  the  squares  of  its  distances  from  the  sides  of  an  equilateral  tri- 
angle is  constant. 

37.  If  a  point  moves  so  that  the  sum  of  the  squares  of  its  distances 
from  any  number  of  fixed  points  is  constant,  show  that  the  locus  will  be 
a  circle. 

38.  Find  the  equation  to  the  circle  when  the  axes  are  inclined  at  an 
angle  oo  =  120°,  the  intercepts  being  (0,  h),  and  (0,  I). 

39.  Required  the  radius  of  the  circle  x'^  +y~+2xycos  oo—hx  —ley  —d. 

|/(A2  +  ^2  —  2111  cos  go) 

Ans.  n — ^ • 

2  sin  03 

40.  Find  the  locus  of  the  middle  points  of  chords  drawn  from  any 
point  of  the  circle. 

41.  Given  the  base  of  a  triangle  and  the  sum  of  the  squares  on  its 
sides  ;  to  find  the  locus  of  its  vertex. 

42.  ABC  is  an  equilateral  triangle,  find  the  locus  of  a  point  P,  such 
that  PA  =  PB  +  PC. 

43.  Find  the  polar  equation  to  the  circle,  the  initial  line  being  a  tan- 
gent. 

44.  Show  that  the  locus 

r  =  A  cos  {(p  —  a)  +B  cos (<p  —  yS)  +  C cos  i<p—  y)  +  etc, 
is  a  circle. 


140  EXAMPLES. 

45.  If  the  centre  be  at  the  pole,  show  that  the  polar  equation  to  the 
chord  of  a  circle  which  subtends  an  angle  2/i  at  the  centre  is  r  =  c  cos  ft 
sec  {q)  —  a)  when  a  is  the  angle  between  the  initial  line  and  the  line  from 
the  centre  which  bisects  the  chord. 

46.  Determine  the  radius  and  centre  of  the  circle  r^  +  2a  (cos  <p  f 
sin  (p)r  —  c*  =  0. 

47.  A  line  is  drawn  from  a  fixed  point  0,  meeting  a  fixed  circle  in  P; 
in  OP  a  point  Q  is  taken  so  that  OP .  OQ  =  Ji^;  find  the  locus  of  Q. 

48.  Find  the  bilinear  equation  to  a  circle,  the  axes  being  a  pair  of 
tangents  and  making  an  angle  of  45°  between  them. 

49.  Find  the  equations  to  the  common  tangents  of  the  two  circles 

x^  +  y^  —  X  —  y  +  4:  =  0,  x'^  +  y^  +  X  +  1/  —  4  =  0. 

50.  If  PQ  be  the  diameter  of  a  circle,  the  polar  of  P  with  respect  to 
any  circle  that  cuts  the  first  at  right  angles,  will  pass  through  Q. 


Of  TJie  Ellipse. 

51.  What  is  the  eccentricity  of  the  ellipse  3.r*  +  4?/*  =  c^  ? 

52.  If  the  normal  to  an  ellipse  at  the  extremity  of  tlie  latus  rectum 
passes  through  one  end  of  the  minor  axis,  show  that  the  eccentricity 

will  be   A/|(i^5  -  1). 

53.  If  the  tangent  to  an  ellipse  at  the  extremity  of  the  latus  rectum 
meets  the  minor  axis  produced  in  iT,  and  tlie  normal  at  the  same  point 
meets  the  major  axis  in  O,  and  if  the  angle  between  the  normal  and  the 
major  axis  is  cot~'  e*,  find  CG  -4-  GK,  C  being  at  the  centre. 

54.  Find  the  locus  of  the  middle  point  of  a  focal  chord. 

55.  Given  the  base  of  a  triangle  and  the  product  of  the  tangents  of 
the  angles  at  the  base ;  prove  that  the  locus  of  the  vertex  is  an  ellipse. 

56.  Show  that  the  equation  to  the  normal  of  an  ellipse  expressed  in 
terms  of  the  tangent  of  the  angle  which  the  line  makes  with  the  axis  of 

««  -  &« 

a;  IS  w  =  mx  —  m    , 

57.  In  the  ellipse  «*  +  4^/*  =  9,  show  that  the  locus  of  the  middle 
points  of  all  chords  passing  through  the  point  (a,  %a)  is  («  —  ^a)*  + 

58.  If  the  ordinate  of  a  circle  whose  equation  is  x^  +  y^  =  r*  be 
increased  in  length  by  the  corresponding  abscissa,  show  that  the  locus 
of  the  extremity  of  the  ordinate  thus  increased  is  an  ellipse. 

59.  Find  the  locus  of  the  intersection  of  a  normal  to  an  ellipse  and  a 
perpendicular  upon  it  from  the  centre. 

Am.  («2  +  y^y  (a^yi  +  ¥x^)  =  c^x^y*. 


EXAMPLES.  141 

60.  Find  the  locus  of  the  intersection  of  a  normal  to  an  ellipse  and  a 
perpendicular  upon  it  from  one  focus. 

Am.  {a'^y^  +  b-x-)  (y^  +  x^  -  cxy  =  c*x^yK 

(This  curve  consists  of  two  loops  one  inside  the  other. — Ed.  Times.  Re- 
print, Vol.  XXIV.,  p.  26.) 

61.  Find  the  locus  of  the  intersection  of  a  perpendicular  from  the 
centre  upon  a  tangent. 

Ans.  p-  =  a^  cos'-^  cp  +  h^  cos^  (p. 

62.  Show  that  the  lengths  of  two  conjugate  semi-diameters,  in  terms 
of  the  eccentric  angle,  are  «'  *  =  a^  cos^  qj  +  b'^  sin'^  ^,  and  h'  '^  =  a'^ 
sin*  tp+b'^  cos^  q>. 

63.  Find  the  locus  of  the  intersection  of  tangents  drawn  through 
the  extremities  of  a  pair  of  conjugate  diameters. 

Ans.  a^y'^  +  ¥  x^  =  2a''¥  {The  Wittenherger,  1878,  p.  40). 

64.  Find  the  locus  of  the  poles  of  the  normals  of  an  ellipse.  (Re- 
print. Ed.  Times,  Lond.,  Vol.  XXVI.,  p.  98.) 

Solution. — A  solution  is  easily  effected  by  means  of  the  eccentric 
angle.     The  equation  to  the  normal  may  be  written,  (Art.  113a), 

a  sec  q)         b  cosec  <p 

5 X  — s —  y  —>■• 

Let  (xi,  yi)  be  the  pole  of  the  normal;  then,  according  to  Article 
150,  equation  (1),  the  equation  of  the  normal  will  be 

«i       yi       1 

-x  +  -^y^l, 
which  must  be  identical  with  the  preceding  equation ; 
Xi      a  sec  (p       y^ 

Eliminating  (p  gives  (dropping  the  subscripts), 

a^y^  +  ¥x^  —  cH'^y^ 

which  is  the  required  equation. 

(Let  the  solution  also  b3  made  by  means  of  the  rectangular  equation  to 
the  normal.) 

65.  If  (a?!,  y ,)  is  the  pole  of  the  chord  P§,  normal  at  P  to  the  conic 
^  +  -T^  =  1,  prove  that  the  equation  to  the  noimal  atQ  is  ^g        g  •  —  + 

-1 ?•  —  =xo     o  .    ,     B.     (Ibid.,  p.  99.) 

[Sua.— The  values  of  a;,  and  y,,  are  given  in  the  preceding  example. 


142  EXAMPLES. 

The  coordinates  of  the  point  Q,  found  by  combining  the  equation  of 
the  normal  PQ  with  that  of  the  ellipse,  will  be 

[ac*  —  ab*  cosec®  q>\  sec  q)         [be*  —  a*h  sec  q)\  cosec  q) 
a*  sec®  q>  +  h*  cosec  q>     '        a*  sec*  q>  ■\-l)^  cosec*  cp    ' 

in  which  substitute  the  values  of  sec  q>  and  cosec  qj  from  the  preceding 
example,  and  find 

then  find  the  equation  of  the  normal  through  this  point.] 

66.  Show  that  the  focal  radii  of  any  point  of  an  ellipse,  in  terms  of 
the  eccentric  angle,  are  r  =  a  (1  —  c  cos  q))  and  r'  =  «  (1  +  e  cos  q»). 

67.  Two  radii  vectors  at  right  angles  to  each  other,  are  drawn  from 
the  centre  of  an  ellipse ;  show  that  the  locus  of  the  intersection  of  tan- 
gents drawn  at  their  extremities  is  an  ellipse  whose  equation  is 

a*      ft*    ~a*  "^  &2  ■ 

68.  If  P  be  any  point  of  an  ellipse,  y  its  ordinate,  A  and  A'  the  ex- 

2&* 

tremities  of  the  maior  axis,  show  that  tan  APA  = 5—. 

■'  ae-y 

69.  The  locus  of  the  middle  point  of  the  normal  to  an  ellipse,  inter- 
cepted between  the  curve  and  the  major  axis,  is  an  ellipse ;  show  that  if 
the  eccentricity  of  the  given  ellipse  be  e,  that  of  the  locus  will  be 

1-:  ' 


(1+6*) 


G' 


70.  Show  that  the  equation  to  the  locus  of  the  middle  points  of  all 
chords  in  an  ellipse  whose  length  is  2?,  is 

aV+6*.r8        a;*         y^ 
*   a^y-'+h'x''^  a^   +  J*    -i-U. 

71.  Find  the  polar  equation  to  the  ellipse,  the  pole  being  at  the  prin- 
cipal vertex,  and  the  major  axis  being  the  initial  line. 

72.  Find  the  polar  equation  to  the  ellipse,  the  pole  being  at  the  point 
(a;n,  yo)  in  reference  to  the  centre,  and  the  initial  line  making  an  angle  a 
with  the  major  axis. 

73.  The  polar  equation  to  the  tangent,  the  pole  being  at  the  left  fo- 
cus, the  major  axis  being  the  initial  line,  and  (p',  q>')  the  tangent  point,  is 

a{\  -  e*) 

P  ~  cos  (^  —  q>')  —  e  cos  g> ' 


EXAMPLES.  143 

74.  Find  the  polar  equation  to  the  curve  given  in  example  60. 

75.  Given  tlie  relative  values  r,  r\  r" ,  of  three  radii  vectors  drawn 
from  the  focus  of  an  ellipse,  and  the  angles  between  them ;  required  the 
relative  value  of  the  major  axis,  the  eccentricity  of  the  ellipse,  and  the 
position  of  the  major  axis. 

Suggestion. — Let  <p  be  tlie  angle  between  the  major  axis  and  r, 
measured  from  the  nearest  vertex,  «the  angle  between  rand  ?•',  and  /3the 
angle  betw^een  r  and  /'.     Then  equation  (5),  page  115,  gives 


1  +  c  cos  cp  ' 
«(1  -  e^) 


r    — 


1  +  ecos(^+  a) ' 

n{l  -  e^) 
1  +  ecos  (<p+  /i)' 


in  which  ?•,  r',  r'\  a,  and  fS  being  known,  «,  e,  and  q)  may  be  found  by 
elimination;  the  last  of  which  gives  the  position  of  the  major  axis  in  ref- 
erence to  r.     We  find 


r  cos  q)  —  r  cos  {(p  +  (i) 

a  (?•  —  r")  —  (r  —  r)  (r  —  ?•"  cos  (3) 

tan  m  =:  — rr-, rr : rj ^—, ttz — : . 

f       ^,   ^y.  —  r)  sm  fi  —  r  (?•  —  r  )  sin  a 

Having  found  cp  from  the  last  equation,  its  value  in  the  preceding 
equation  gives  e,  and  from  these  a  may  I)e  found  in  terms  of  r. 

[Remark. — By  observing  the  apparent  diameter  of  the  sun  on  three  dif- 
ferent dates,  and  the  angles  passed  over  between  these  dates,  the  eccentri- 
city of  the  earth's  orbit,  the  position  of  its  major  axis,  and  the  relative  value 
of  the  major  axis  may  be  found.  It  is  difficult  to  observe  the  diameter  of 
the  sun,  so  it  is  better  to  observe  the  apparent  angular  velocity,  and  use  the 
principle  that.  The  relative  distances  of  the  sun  are  to  each  other  as  the 
aqtuire  roots  of  the  sun's  apparent  angular  velocity.] 

The  Parabola. 

76.  A  circle  radius  r  touches  a  parabola  whose  latus  rectum  is  Ap  at 
the  vertex,  show  that  if  it  cuts  the  parabola,  the  distance  from  the  ver- 
tex to  the  straight  line  joining  the  points  of  intersection  will  be  2r  —  ip. 

77.  Find  the  equation  to  the  axis  of  the  parabola  whose  equation  is 
(«  — a)*=a(a;  -i-  y). 

78.  Show  that  the  vertex  of  the  parabola  (3a!  —  4^)*  -  50ax  +25a'  = 


),  is  (?«,_?«). 


144  EXAMPLES. 

79.  QOQ  '  is  a  fixed  chord  in  a  parabola,  P' OP  another  chord  paral- 
lel to  a  given  straight  line.  Op  is  taken  on  OP^  produced  if  necessary,  such 
that  OP .  Op  —  OQ  .  00'  \  show  that  the  locus  of  j;;  is  a  parabola. 

80.  A  normal  is  drawn  at  the  point  {x\  y  ) ;  find  the  point  where  it 
again  cuts  the  parabola. 

81.  Show  that  the  equation  to  the  parabola  referred  to  a  pair  of  tan- 
gents at  the  extremities  of  the  latus  rectum  is  /y/ic  -h  '^y  =  ^ p  ^g  . 

82.  From  any  point  there  can  no  more  than  three  normals  be  drawn. 

83.  Find  the  points  of  intersection  of  the  parabolas  y^  —  2px,  and 
a-z  +  ipy  =  0. 

84.  Through  the  point  («',  y' )  on  a  parabola  a  normal  is  drawn ;  find 
the  equation  of  a  tangent  parallel  to  the  normal,  and  the  point  of  contact 
of  the  tangent. 

85.  Find  the  equation  to  the  normal  passing  through  the  point  of 
contact  of  the  tangent  in  the  preceding  example. 

86.  Find  the  locus  of  the  middle  points  of  focal  chords  to  a  pa- 
rabola. 

87.  Show  that  the  locus  of  the  centre  of  a  circle  which  touches  a 
given  line  and  given  circle  is  a  parabola. 

88.  Find  the  locus  of  the  centre  of  a  circle  inscribed  in  a  variable 
sector  of  a  given  circle,  one  of  the  bounding  radii  being  fixed. 

89.  Find  the  locus  of  the  intersection  of  mutually  perpendicular  nor 
mals  to  a  parabola. 

(Solution. — The  equations  to  the  normals  will  be 
y-y  =  -- ^(sB-a;'), 

y- y"=--|-(aj-a!"). 
Because  they  are  mutually  perpendicular,  we  have 

r  =  -^,  (1) 

and  the  equation  to  the  curve  gives 

v-J'^  ^"-lll. 

"  -  2p'  ^   -  %p  ' 

and  the  equations  become,  by  substitution,  clearing  of  fractions,  and 
multiplying  the  first  by  2y, 

<i,yy'i  _  4pj,y'  X  =  4:p^y^  -  4:p'yy\ 
2yy'^  +  2p^y'^  =  2pxy'^  —  p*^. 


EXAMPLES.  145 

Subtracting  and  solving  for  y',  we  have 


This  gives  two  points  from  which  normals  may  be  drawn,  and  these 
normals  must  be  mutually  perpendicular,  so  that  if  one  be  y'  the  other 
will  be  y"  and  we  have  from  (1) 


/ 


2(i>  -  X) 


-Vy'-' 


which  reduced  gives 


y''  =hp{x-%p); 


hence  the  locus  is  a  parabola  having  one-fourth  the  parameter  of  the  given 
curve,  and  whose  vertex  is  on  the  axis  of  the  given  curve  at  a  distance 
from  that  vertex  equal  to  three  times  the  distance  of  the  given  focus 
from  the  vertex  of  the  given  cui-ve.) 

90.  Prove  that  the  equation  to  the  normal  in  terms  of  the  tangent  of 
the  angle  which  it  makes  with  the  axis  of  the  cune,  is,  y  =  inx  —  pm  — 

(0b8. — The  preceding  example  may  be  solved  by  means  of  this  equa- 
tion.) 

91.  If  two  parabolas  whose  axes  are  mutually  perpendicular  inter- 
sect in  four  points,  prove  that  the  four  points  are  in  the  circumference 
of  a  circle. 

92.  If  two  parabolas,  having  a  common  vertex,  and  axes  at  right 
angles  to  each  other,  intersect  in  the  point  x'y  ;  then 

2p  :  x  ::  y   :  2p'. 

[Obs. — This  property  enables  one  to  insert  graphically  two  geometric 
means  between  two  given  lines,  and  for  this  reason  has  been  used  in  solv- 
ing the  celebrated  problem  of  the  Duplication  of  the  Cube.  ] 

93.  Parabolas  have  their  axes  parallel  and  all  pass  through  two  given 
points ;  prove  that  their  foci  lie  in  a  conic  section.  (Tait  &  Ivelland's 
Q^Mtem^o?^s.) 

94.  Prove  that  the  locus  of  the  poles  of  normals  to  a  parabola  is 
{x+p)y'+{p»=0.    {EduMtional  Times,  Reprint,  1876,  Vol.  XXVI.,  p.  99.) 

95.  An  ellipse  and  a  parabola  have  a  common  focus,  and  the  other 
focus  of  the  ellipse  moves  on  the  directrix  of  the  parabola ;  show  that 
the  points  of  contact  of  a  common  tangent  subtend  a  right  angle  at  the 
common  focus.     {Math.  Visitor,  1878,  p.  45.) 

10 


146 


EXAMPLES. 


General  Equatiorts. 

96.  The  equation 

y^  —  ^xy  +  a;2  —  1  =  0, 
is  the  locus  of  what  curve? 


/I 


Ite.  143. 
y  97.  Show  that  the  equation 


y 


JAB 

F 


y^  —  2xy  +  x^  +  2y-2x+  l  =  Oy 


Via.  m.      ^  *^®  locus  of  a  single  right  line. 
Y 


98.  The  equation 

y^  —2xy  —  x^  —  2x  —  2  =  0, 

is  the  locus  of  what?  Does  it  cut  either 
or  both  the  coordinate  axes?  Wliat  is 
the  inclination  of  the  diameter  with  the 
axis  of  «? 


99.  y*  +  y  +  1  =  0,   is  the  locus   of 
what? 

g  (AB  is  the  diameter  and  the  lines  im- 

aginary.) 


X         100.  Determine  the  character  and  posi- 
tion of  the  locus 

y^  -2x'>  +2y  +1  =  0. 

Is  it  symmetrical  in  reference  to  the  axis  of  y? 


EXAMPLES. 


147 


101.  The  equation 

y^  —  2i'y  +  2)2  4-  a:  =  0, 

is  the  locus  of  wliat  ? 

{AB  la  a  diameter  whose  equation  \sy  —  x.) 

102.  The  equation 

2/2  _  2xy  +  x^  +  2y  +1  =  0, 

is  the  locus  of  what  curve?  Is  the  curve  tangent 
to  the  axis  of  y'i  What  is  the  slope  of  the  di- 
ameter? 


103.  The  equation 

y^  —  2xy  +  a;2  -  2y  —  1  =  0, 
b  the  locus  of  what  curve? 


104.  What  locus  is  represent- 
ed by  the  equation 

y^  —  x^  +2x  -  2y  +  1  =  0  ? 

Does  it  cut  the  axis  oi  y'i.  Does 
it  cut  the  axis  of  a;  ?  Has  it  a 
centre  ? 


Fig.  150. 


Fig.  151. 


105.  If  A  and  B  are  fixed  points  in  the  axes  Ox,  Oy,  and  a,  &,  are 
always  taken  on  Ox,  Oy,  so  that 


^-^varies  inversely  as -^ 


OB  ' 


show  that  the  locus  of  the  intersection  of  Al)  and  Eb  is  a  conic  section 
touching  Ox,  Oy,  in  J.  and  .8. 


CHAPTER  YI. 

LOCI     IN     SPACE. 

Of  the  Point,  Right  Line  and  Plane, 

188.  The  loci  previously  considered  have  been  con- 
fined to  one  plane,  and  the  coordinate  axes  have  been  taken 
in  that  plane.  When  loci  are  referred  to  three  intersecting 
planes,  they  are  called  loci  in  spacco 

189.  Definitions.  —  Coordinate  planes  are  three  planes 
intersecting  each  other,  to  which  loci  are  referred.  The 
three  planes  will  intersect  in  a  common  point,  and  the 
planes  taken  two  and  two  will  intersect  in  right  lines  passing 
through  the  common  point.     Coordinate  a^ces  are  the  lines  of 

intersection  of  the  coordinate  planes, 
as  OX,  0  Y,  and  OZ.  The  origin  0  is 
the  common  point  of  intersection  of 
the  coordinate  axes.  Rectangidar  coor- 
dinates are  those  in  which  the  coordi- 
nate axes  make  right  angles  with 
each  other.  All  coordinates  not  rect- 
angular are  oblique.  The  coordinate 
planes  are  assumed  to  be  indefinite  in 
extent,  and  divide  all  space  into  eight  parts  or  solid  angles, 
and  if  the  axes  are  rectangular,  the  eight  parts  will  be  equal 
to  each  other.  The  plane  YOX  is  generally  assumed  to  be 
horizontal,  and  the  other  two,  vertical,  but  as  they  may  have 
any  position  in  space,  this  assumption  is  entirely  arbitrary. 

190.  Axis  of  a  Plane. — Any  line  perpendicular  to  a 
plane  may  be  considered  as  the  axis  of  that  plane.     "When 

148 


191,  193.J 


THE  POINT. 


149 


the  axes  are  rectangular,  the  co(>rdinate  axis  OZ  is  the  axis 
of  the  plane  YOX,  since  it  is  perpendicular  to  that  plane, 
and  similarly  for  the  other  planes. 

191.  Order  of  the  Angles.  — The  angle  Z-YOX, 
which  is  above  the  horizontal  plane  YOX,  at  the  right  of  the 
vertical  plane  ZO  Y,  and  in  front  of  the  vertical  plane  -^OX, 
is  called  the  first  angle.     Similarly, 

Z-XOY'    is  called  the  2d  angle, 

Z-Y'OX''      "        "    3d      " 

Z-X'OY  "      "        "    4th     " 

-Z-YOX    "      "        "    5th     "     ; 
and  so  on. 

In  analysis,  however,  the  angles  are  determined  by  the 
signs  of  the  coordinates.  Thus,  OX  is  plus  x,  OX',  minus  a?; 
OZ,  plus  z,  OZ',  minus  2  ;  0  Y,  plus  y,  and  0  Y  minus  y. 


The  Point 

192.  Coordinates  of  a  Point. — The  position  of  a  point 
is  determined  by  its  distance  from  the  coordinate  planes 
measured  on  lines  parallel  to  the  coordinate  axes.  Let  p  be 
the  point.  Through  p 
draw  pG  parallel  to 
the  axis  of  x,  and  let  6^ 
be  the  point  where  it 
intersects  the  plane 
ZOF;  then  will  ^(?  be 
the  X- ordinate  of  the 
point.  Similarly,  pE 
is  the  y-ordinnte,  and 
pD  the  z  -  ordinate. 
Three  coiirdinates  de- 
termine a  point;  for 
three  right  lines  can  intersect  in  one  point  only.  If  the  co- 
ordinates are  rectangular,  they  will  be  perpendicular  to  the 
respective  planes,  that  is,  pG  will  be  perpendicular  to  the 
plane  ZO  Y,  and  similarly  for  the  others. 


-' 

^-y^^ 

—  X 

0. 

p  ,.'■'' 

E    ,. 

F 

—  Y 

^^ 

^ 

^^ 

Y 

^. 

-Z 

Fig.  153. 


150  LOCI  IN  SPACE.  [192a,  192&. 

Tlie  coordinates  of  a  point  may  be  expressed  algebraic- 
ally by  the  equations 

X  =  a,        y=h        z  =  c, 

and  these  are  called  equations  to  the  point.     Three  determi- 
nate equations  of  the  first  degree  are  therefore  necessary  and 
sufficient  for  determining  a  point  in  space. 
We  have,  for  a  point  in  the 

first  angle  x=  +  a,  y  =  +  b,  z=  +  c, 

second  angle  x=  +  a,  y  =  —h,  z  =  +  c, 

third  angle  x  =  —a,  y  =  —  b,  z=  +  c, 

fourth  angle  x=  —  a,  y  =  +  b,  z  =  +c, 

fifth  angle  x—  +  a,  y  =  +  b,  z=  —  c, 

sixth  angle  x—  +  a,  y  =  —b,  z=  —  c, 

seventh  angle  x—  —  a,  y  =  —  b,  z=  —c, 

eighth  angle  x=  —  a,  y  =  +  b,  'z=  —  c. 

192a.   Coordinates  of  particular  Points. — For  a  point  at  the 
origin,  we  have 

x  =  0,        y  =  0,        z  =  0. 

For  a  point  on  the  axis  of  x,  we  have 

x=  ±  a,        y  =  0,        z  =  0, 

and  similarly  for  a  point  on  any  other  axis. 
For  a  point  in  the  plane  xy,  we  have 

x=  ±  a,        y  =  ±b,        2  =  0, 

and  similarly  for  a  point  in  any  other  coordinate  plane. 
192b.  Equations  of  the  Axes. — ^For  the  axis  of  a;,  we  have 

2  =  0,         y  =  0,        and  X  indeterminate. 

The  value  of  x  being  indeterminate,  will  represent  every 
point  of  the  axis  in  succession. 
For  the  axis  of  y,  we  have 

2  =  0,        X  =0,        and  y  indeterminate; 


193-195]  THE  RIGHT  LINE.  151 

and  for  the  axis  of  z, 

ic  =  0,        y  =0,        and  z  indeterminate, 

193.  Distance  bet^ween  t-wo  Points. — Let  the  coordi- 
nates of  one  point  p,  be  x^,  y^,  z^,  and  of  the  other  point  0, 
be  iCo ,  ?/2 ,  2  2 ;  then  will  the  distance  O/j  between  the  points 
be  the  diagonal  of  a  parallelopiped,  hence 


and 


Op  =  l=  V{Xi  -  x.^-'  +  (y,  -  y,y  +  {z,  -  z,)\ 
If  the  point  0  be  at  the  origin,  then  iCj  :=  0,  ?/2  =  0,  «2  =  0, 


l  =  Vxi-  +  y;'+z^\ 


"which  is  the  distance  of  any  point  from  the  origin. 

194.  Projections. — The  foot  of  a  perpendicular  from 
a  point  upon  a  plane  ^  ^ 
is  the  projection  of  the 
point  on  that  plane. 
Thus  D  is  the  projec- 
tion of  the  point  P 
upon  the  plane  xy  ;  E, 
upon  xz  ;  G,  upon  zy. 
The  lines  PD,  PE, 
PG,  passing  through 
the  point  perpendicu- 
lar to  the  respective 
planes  are  called  pro- 
jecting lines,  or  lines  of  projection. 

The  projection  of  a  line  is  the  projection  of  every  point  of 
the  line.  This  is  true  of  curved  as  well  as  of  right  lines. 
If  a  curved  line  be  in  the  plane  of  two  projecting  lines,  its 
projection  will  be  a  right  line,  but  in  other  cases  its  projec- 
tion will  be  curved. 

The  Right  Line. 

195.  Equations  to  a  Right  Line. — A  line  is  given  by 
its  projections.     For,  if  perpendiculars  be  erected  from  the 


G  p-'-'— 

^       1 

■"• ; 

E     ,. 

-y 

—  X 

a 

F 

^       X 

Hy^. 

0    /^ 

^^ 

Y 

-Z 

Fig.  154. 


152 


LOCI  IN  SPACE. 


[196. 


corresponding  points  of  the  pro- 
jections, they  will  be  the  project- 
ing lines.  But  the  projecting  lines 
intersect  in  space  on  the  given 
line,  and  hence  determine  its  po- 
sition. 

Let  ABhQ  the  line  in  space. 
The  projection  of  the  point  B  on 
the  plane  zy  is  D;  similarly,  G 
is  the  projection  of  A;  and  CD 
will  be  the  projection  of  the  given  line  on  the  plane  zy. 
Also,  AE  will  be  its  projection  on  zx ;  and  FB,  its  projec- 
tion on  xy. 

Let  m  be  the  tangent  of  the  angle  which  the  line  AE 
makes  with  the  axis  of  z,  and  n  that  of  CD  with  the  same 
axis.  Also  let  a  =  OE  the  intercept  of  AE  on  the  axis  of  x, 
and  h  —  OD  the  intercept  of  CD  on  the  axis  of  y.  Then, 
according  to  Article  28,  we  have  for  the  equation  of  AE^ 

x  =  mz  +  a,  (1) 


Pig.  155. 


and  for  CD, 


y  =nz  +  b; 


(2) 


which  are  the  required  equations.     Two  equations  of  the 
first  degree,  each  having  two  of  the  required  variables,  are 
necessary  and  sufficient  for  determining  a  right  line  in  space. 
Eliminating  z  from  these  equations  gives 


X  —  a  _  y  —  h 


(3) 


which  is  the  equation  of  FB,  the  projection  of  the  given  line 
on  the  plane  xy. 

196.  Intersection  of  t"wo  right  lines  in  space. — Let 
the  lines  be 

x=mz+a,  and  x  =  m'z  +a', 

y  —  nz  +  b,  y  =  n'z  +  b'.  ^  ^ 

Considering  the  equations  as  simultaneous,  and  eliminat- 
ing x  and  y,  we  have 


197.]  THE  EIGHT  LINE.  153 

{m  —  m') z  =  a'  ~  a;  (n  —  n')z=  b'  —  b 

* 

a'  —  a         ,  b'  —  b  ,n\ 

.'.  z  = r  and  z  = -.  (z) 

m  —  m  n —w 

Eliminating  z  between  equations  (2),  gives 
a'  —  a        b'  —  b 


m  —  m        n  —  n 


(3) 


"whicli  establishes  a  relation  between  the  constants  of  the 
equations,  and  is  an  equation  of  condition,  (Art.  39).  Hence 
we  infer  that,  if  the  relation  between  the  eight  constants  of 
equations  (1)  gives  equation  (3),  the  lines  will  intersect,  but 
otherwise  they  will  not  intersect.  Lines  in  space  which  are 
not  parallel  may,  therefore,  have  an  infinite  number  of  posi- 
tions and  not  intersect.  If  equation  (3)  is  true,  either  of 
equations  (2)  will  give  the  ordinate  z  of  the  intersection,  and 
this  value  in  (1)  gives  for  the  other  coordinates 

ma'  —  m'a  nb'—n'b 

x  = J—',         y  = T-. 

m  —  m  n  —  n 

If  m  =  m,  equation  (3)  shows  that  n  —  n',  and  we  have 

a;  =  00  ;         y  =  a:);         z  —  co; 

and  the  lines  are  parallel ;  and  the  equations  m  =m'  and 
n  =  n'  are  tJw  equations  of  condition  of  the  parallelism  of  two 
right  lines  in  space. 

197.  Equation  to  a  line  -which  passes  through  two 
points  in  space. — Let  {x',  y\  z)  be  one  point,  and  {x",  y",  z'), 
the  other.     Then  proceeding  as  in  Article  40,  we  have 

,      x"—  x   ,         ,v  ,      y"—  y'  ,         ,.        ^^. 

x-x  =~rr~,  {^-^);      y-y=-^-r, — fr(^-0;    (i) 

which  are  the  required  equations.  If  the  line  passes  through 
one  point  only,  the  equations  become 

X  —x'  =  m(z  —  z'),        y  —y'  =n{z  —  z');        (2) 

in  which  m  and  n  are  indeterminate. 


154 


LOCI  IN  SPACE. 


[198,  199. 


198.  Inclination  of  a  line  to  the  three  Axes. — Draw 
a  line  through  the  origin  parallel  to  the  given  line ;  it  will 
make  the  same  angles  with  the  axes 
as  those  of  the  given  line.     Let  the 
equations  of  the  line  be 


X  =  mz,        y  =  nz. 


(1) 


Let  P  be  any  point  (x,  y,  z)  of  the 
line,  and 

Y=POY,        Z=POZ,        r=OP. 

Then    x  =  r  cos  X;        y  =  f  cos  Y;        z  =  r  cos  Z.      (2) 

Squaring  these,  and  substituting  in  the  equation 

7^  =  ay^  +  y^  +  z\ 

gives  r^  =  r^  cos*  X  +  r^  cos^  Y  +  7^  cos^  Z; 

.'.  cos^ X  +  cos^  Y  +  cos' Z=l;  (3) 

by  means  of  which,  if  two  angles  be  given,  the  third  may  be 
found. 

Substituting  the  values  of  x  and  y  from  equation  (1)  in 
the  value  of  r'  gives 


}?  = 


7^  =  m V  +  wV  +  z^ 


1  +  n^  +  m^ 
.  cobZ=  ± 

m 


r'cos^Z  (from  (2)); 
1 


cos  r± 


(4) 


(5) 


Similarly 

cos  X=  ±  — 7 , 

Vl  +  n^  +  m^  '  Vl  +  n^  +  m? 

by  means  of  which  the  required  angles  may  be  found  when 
the  equations  to  the  lines  are  given. 

189.  Angle  between  two  lines. — The  angle  between 


199] 


THE  RIGHT  LINE. 


155 


two  lines  in  space,  whether  the  lines  intersect  or  not,  is  the 
same  as  the  angle  between  two  lines 
drawn  through  the  origin  parallel  re- 
spectively to  the  given  lines.  Let  AP 
be  parallel  to  one  line,  and  A  Q  parallel 
to  the  other,  then  will  the  angle  PA  Q  be 
the  required  angle.  Since  the  angle  will 
be  independent  of  the  length  of  the 
lines,  take  each  equal  to  unity,  and 
let  PQ  =  d,  and  PAQ  =0.  Then,  from  Trigonometry,  we 
have 


Fig.  157. 


.     AP'  +  AQ--d-      2-d'' 
"^^^"        2AP   AQ       =-^ 


(1) 


Let  the  points   P  and  Q  be  respectively  {x,  y',  z')  and 
ix",  y",  z") ;  then 


d'  =  {x"-x'f  +  {y"  -  yj  +  {z"  -  zy. 


(2) 


But  from  equations  (2)  of  Article  198,  since  r  =  1,  we 
have 

a;'  =  cosX',        y'  =  co^Y',        z'  =  cosZ', 
x"  =  cos  X",       y"  =  cos  Y",     z"  =  cos  Z' 


'■',}      (3) 


These  values  substituted  in  equation  (2)  give 
d^  =  {cos'X'+  cos-Y'  +  cos,^Z')  +  (cos'X"  +  cos'Y"+  cos'Z") 
-2(cosX'cosX"+  cos  F'cos  Y'  +  cosZ' cosZ").     (4) 

Reducing  this  by  means  of  equation  (3)  of  Article  198,  sub- 
stituting in  (1)  above,  and  reducing  gives 

coB^=  cosX'  cosX"  +  cos  Y'  cos  Y"  +  cos-^'cos^",     (5) 

which  is  one  form  of  the  required  value. 

Substituting  in  this  equation  the  values  corresponding  to 
equations  (4)  and  (5)  of  the  preceding  Article,  we  have 


coa6=  ± 


mm  +  nn'  +  1 


^[{m^  +  n'  +  l){m"'  +  n"  +  1)] ' 


(6) 


156 


LOCI  m  SPACE. 


[200,  201 


If  the  lines  are  mutually  perpendicular,  cos  (9  =  0,  and 
we  have 

mm!  +  nn'  +1  =  0, 

which  is  the  equation  of  condition  of  mutually  perpendicular  lines 
in  space. 

If  the  lines  are  parallel,  cos  ^  =  1,  and  squaring  both 
members,  transposing  and  reducing,  gives 

(m  —  m')-  +  {n  —  n'Y  +  {mn'  —m'nf  =  0; 

and  since  each  term  is  a  square,  they  must  separately  be 
equal  to  zero,  or 


ni  —  m. 


n  —  n 


mn  =  mn, 


the  third  of  which  results  from  the  other  two.     This  result 
agrees  with  that  found  in  Article  196. 


The  Plane. 

200.  A  Plane  may  be  generated  by  a  right  line  moving 
along  another  right  line  and  remaining  constantly  parallel 

to  its  first  position.  Let 
ABC  be  a  plane  supposed 
to  be  generated  by  the  line 
DE  moving  along  the  line 
BC  and  being  constantly 
parallel  to  AB.  The  equa- 
tion of  every  point  of  the 
moving  line  in  every  posi- 
tion will  be  the  equation 
of  the  plane.  The  trace  of 
a  plane  is  its  intersection 
with  one  of  the  coordinate  planes. 

201.  Equation  to  a  Plane  in  terms  of  its  Inter- 
cepts.— ^Let  a,  h,  c,  be  the  intercepts  on  the  axes  re,  y,  z,  re- 
spectively. The  equation  of  the  line  AB  will  be,  (Eq.  (5), 
Art.  29), 

X     y      ., 
-  +  i  -1. 
a     0 


Fig.  158. 


202]  THE  PLANE.  157 

Let  FG  be  the  projection  on  the  plane  xy  of  the  generating 
line  DE,  and  let  the  intercepts  of  this  projection  he  a  —  OG 
and  b'  —  OF,  then  will  its  equation  be 

a      0 
But  from  the  similar  triangles  B  OA  and  G  OF,  we  have 


a      b 
a 


b  b       ab 


which,  substituted  in  the  preceding  equation,  gives 

X     II      a 
aba 

But  the  ordinate  of  the  line  DE  is  z,  and  the  similar  tri- 
angles BOG,  BGE,  give 

c  _       z       _     .  ^'  _  1       ^  . 


a     a  —  a  a  c 

and  the  preceding  equation  becomes 

X      y      z      ^ 
-  +  f  +  -  =  1, 
a      b      c 

which  is  the  required  equation. 

202.  Every  Equation  of  the  First  Degree  between 
three  variables  may  represent  a  plane. — The  equation 
will  be  in  the  form 

Ax  +  By  +  Cz-\-  D  =  Q, 

which  becomes,  by  dividing  through  by  D, 

But  whatever  be  the  values  of  A,  B,  C,  D,  the  coefficients  of 

X,  y,  z,  may  be  represented  by >  "~  a»  ~  "'  ^^^  *^®  equa- 

a       0       c 


158 


LOCI  m  SPACE. 


[203. 


tion  be  thus  reduced  to  tliat  of  the  last  equation  of  the  pre- 
ceding Article,  which  equation  is  that  of  a  plane. 

203.  Discussion  of  the  Equation  Ax+By  +  Cz  +  I)=0. 
Let  2  =  0,  then  the  equation  becomes  Ax  +  By  +  D  =  0, 
which  is  the  equation  of  the  trace  of  the  plane  on  the  coor- 
dinate plane  ,/y.  Similarly,  Ax  +  Cz  +B  —  0  is,  the  equation 
of  the  trace  on  X2,  and  By  +Cz  -f  i>  =  0  is  the  trace  on  yz. 

D 


If  2=0,  and  y  =  0,  then  x 


A 


,  which  is  the  abscissa 


OB  of  the  point  where  the  plane  cuts  the  axis  of  x.   Similarly, 
y  =  —  —  gives  the  point  A  where  it  cuts  the  axis  of  y ;  and 


2  = 


D 

C 


gives  the  point  0  where  it  cuts  the  axis  of  z. 


If  the  plane  does  not  cut  the  axis  of  z,  then  the  intercept 


on  that  axis  will  be  infinite,  or  ^^  =  oo ; 
terminate,  we  have 


C  =  0,  and  z  inde- 


Ax  +  By  +  0.Z  +  D  =  0 

for  the  equation  of  a  plane  parallel  to  the  axis  of  z  and 
oblique  to  both  the  other  axes. 

If  the  plane  is  also  parallel  to  the 
axis  of  y,  we  have 

Ax  +  0.y  +  0.z+D  =  0. 

It  cannot  be  parallel  to  the  three 
axes  at  the  same  time ;  for  the  sum 
of  three  zeros  cannot  equal  a  finite 
quantity. 

If  the  plane   passes  through  the 
origin,  we  have  for  that  point  x  =  0, 
2/  =  0,  2  =  0 ;  .  •.  D  =0,  that  is,  the  ahsolnte  term  will  he  zero. 

If  the  plane  coincides  with  the  coordinate  plane  zy,  it 
will  pass  through  the  origin,  hence  B=0,  and  x  =  0,  and 
the  equation  becomes 


Fig.  159. 


O.y  +  0.z  =  0, 


204,  205.]  THE  PLANE.  159 

whicli  is  tlie  equation  of  the  coordinate  plane  yz.    Similarly, 
for  the  other  coordinate  planes. 

204.  Equation  to  a  Plane  in  terms  of  the  perpen- 
dicular from  the  origin  upon  the  plane  and  its  direc- 
tion-cosines.— Let^  be  the  perpendicular,  and  X,  Y,  Z,  the 
angles  which  it  makes  with  the  axes  x,  y,  z,  respectively. 
These  angles  will  be  the  direction-angles,  and  cos  A',  etc.,  the 
direction-cosines.  Also,  let  a,  b,  c,  be  the  intercepts  of  the 
plane  on  the  axes.     Then, 

aGosX  =  p,         h  cos  Y=p,        c  cos  Z  =  p,      (1) 

Substituting  the  values  of  a,  h,  c,  found  from  these  equations, 
in  the  last  one  of  Article  201,  gives 

X  cos  X  +  y  cos  Y  +  z  cos  Z  =^,  (2) 

which  is  the  required  equation. 

205.  To  reduce  Ax  +  By  +  Cz  +  I)  =  0  to  the  form  last 

found. — Dividing  this  equation  by  B  and  the  former  hjp 

give 

A        B        C       .      ^ 
^a^+^y +  ^2  +  1  =  0, 

cos  X        cos  Y         cos  Z       ^      ^ 

X— y z  +  1  =  0; 

p  p  p 

.-.  cosX=  -A^;  C0&  Y=-B^;  cobZ=-  C^. 

Squaring  and  adding,  gives 

p  _        ± 

B~  ^  VA'  +  B^+C  ' 

Therefore,  if  the  coefficients  of  the  given  equation  be  di- 
vided by  V  ^^  +  B^  +  C^,  the  resulting  coefficients  will  be 
respectively  the  values  of  cos  X,  cos  Y,  cos  Z,  and  p.  The 
sign  of  the  radical  must  be  such  as  to  make  p  always  posi- 
tive, which  sign  will  determine  the  essential  signs  of  cos  X, 
cos  F,  cos  Z. 


160  LOCI  IN  SPACE.  [206,  207. 

206.  Intersection  of  Two  Planes. — Let  the  equations 
to  the  planes  be 

Ax  +  By  +  Cz  +D  =0\ 

and  A'x+B'y  +  C'z  +  D'=0. 

Eliminating  z  gives 

{AC'-A'C)x  +  {BC'-B'C)y  +  DC'-D'C  =  0;     (1) 

which  is  the  equation  of  the  projection  of  the  line  of  inter- 
section on  the  coordinate  plane  xy.  Similarly,  the  equation 
of  the  projection  on  the  plane  xz,  will  be 

{AB'  -A'B)x+  {B'C  -BG')z  +  B'D-BD'  =  0;    (2) 

and  on  yz, 

{A'B  -AB')y  +  {A'G-  AC')z  +A'D  -  AD'  =0.        (3) 

If  the  planes  are  parallel  to  each  other,  ic  =  oo ,  2/  =  oo , 
2  =  00 ,  and  the  coefficients  of  x,  y,  z  must  be  zero ; 

,:AB'  =  A'B,        AC'  =  A'C,        BC  =  B'C',  (4:) 

which  are  the  equations  of  condition  of  the  jparaUelism  of  two 
ptaTies. 

207.  Equation  to  a  Plane  passing  through  three 
Points. — Let  the  points  be  {x,  y',  z'),{x",y",z"),  {x",y"',z"'). 
Substitute  these  successively  for  x,  y,  z,  in  the  equation 

Ax  +  By+  Cz  +  I)  =  0, 

and  find  the  values  of  A,  B,  C,  in  terms  of  x\  x",  etc.,  and 
resubstitute  the  values  of  A,  B,  C,  in  the  equation.  In  this 
way  we  may  find 

[y'{z"-z"')+y"(z"'-z') +y'"{z'-z")]x  \  r  (y"z"'-y"'z")x' 
+  [z'ix"  —x'")  +z"{x"'—x')  +  z"'{x'—x"j]i/  >■  =  <  +iy"'z'  —y'z"')x" 
+  [x\y"-y"')  +x"iy"'  -y')  +x"'{y'-y")]z  '        '  +(y'z"  -y"z')x"', 

for  the  required  equation. 


208,  209.]  THE  PLANE.  161 

208.  Inclination  of  a  Plane  to  the  Coordinate 
Planes. — The  angle  required  is  the  angle  between  two  per- 
pendiculars from  the  origin  to  the  respective  planes.  The 
angle  between  the  given  plane  and  the  plane  xy  will  be  the 
angle  between  the  axis  of  z  and  a  perpendicular  from  the 
origin  to  the  given  plane.  Letting  Z  be  this  angle,  we  have, 
(Art.  205), 

Similarly,  for  the  inclination  to  the  planes  zy  and  sx  re- 
spectively, we  have 

cosX=  ±   ,,  .2  ■  P2 ,  ^ox ;   cosr=± 


209.  Angle  between  two   Planes. — Let  X',   Y',  Z', 

and  X",  Y",  Z",  be  the  angles  between  the  coordinate  axes 
and  the  respective  perpendiculars  from  the  origin  upon  the 
planes,  and  cp  the  angles  between  the  planes.  Then,  fromi 
Article  199,  we  have 

cos  q)  =  cos  X'  cos  X"  +  cos  F'cos  Y"  +  cos^'  cosZ  ";     (1) 

in  which  substitute  the  values  of  cos  X  \  etc.,  (Art.  208),  and 
we  have 

-  4-  AA'  +  BE'  +  CC'  ,o^ 

.       '^''^'^~  ^  ^[{A^  +  B^+G'){A'^  +  B'^+G")y        ^  ' 

which  is  the  required  equation. 

If  the  two  planes  are  perpendicular  to  each  other,  cos  cp 
=  0,  and  we  have 

AA'  +  BB'  -i-  CC'  =  0,  (3) 

which  is  the  equation  of  condition  for  the  mutual  'perpendicu- 
larity of  two  planes. 

[If  the  two  planes  are  parallel,  cos  q)  =  l  and  we  may  find 

AB=A'B,        AC'=A'C,       BG'  =  B'G,  (4) 

as  found  in  Article  206. 
11 


162  LOCI  IN  SPACE.  [210-212. 

If  the  second  plane  is  parallel  to  the  plane  xy,  the  angle  between  them 
will  be  the  same  as  between  the  first  plane  and  the  plane  xy  ;  and  we  will 
have,  (Art.  203),  A'  —  Q,  B'  =  0,  and  equation  (3)  becomes 

G  -^ 

costpxy-  ^(A'^+B'^  +  a^y 
Similarly,  .    (5) 

B  A 

cos  <pxz  =    ^(^2+J58^C'8)  ;       COS    <py«  -    ^(^S  +^8  +  Q  8)' 

88  found  in  the  preceding  Article.] 

310.  Equation  to  a  Plane  parallel  to  Ax  -\-  By  +  Cz 
+  j)^  0.— It  will  be  of  tlie  form  Ax  +  B'y  +  C'z  +  D'=0. 
Substitute  the  values  of  A '  and  C ',  deduced  from  equation 
(4)  of  Article  206,  and  we  have 

^  Ax  +  By  +  Cz  +  ^D'=0; 

or,  the  eqvutions  of  paraEd  planes  differ  ordy  in  their  absolute 
terms. 

211.  Equations  to  a  Plane  perpendicular  to  Ax  +  By 
+  Cz  +  D  =  0. — The  equation  will  be  of  the  form  of  A  'x 
+  B'y+  G'z  +  D'=0.  Substitute  the  value  of  A ' from  equa- 
tion (3)  (Art.  209),  and  we  have 

_BB'  +  CC'^  +  B'y  +  C'z+D'=0, 

for  the  required  equation.  But  since  this  equation  contains 
the  undetermined  constants  B'  CD',  there  may  be  an  infi- 
nite number  of  planes  passed  perpendicular  to  the  given 
one. 

Line  and  Plane. 

2 IS.  To  find  the  Point  where  a  Line  pierces  a 
Plane. — Let  Ax  +  By  +  (7^  +  Z)  =  0  be  the  plane,  and  x  = 
mz  +  a,y  =nz  +  b,  the  line.  Substitute  the  values  of  x  and  y 
from  the  equations  of  the  line  in  the  equation  to  the  plane 
and  find  z.     This  gives 

_  _  aA  +  bB  +  I) 
~      niA  +  nB  +C* 
for  the  s  -  coordinate. 

Li  a  similar  manner  the  other  coordinates  may  be  found. 


218.] 


LmE  AND  PLANE. 


163 


If  tlie  line  is  parallel  to  the  plane,  then  z  =  <X) ,  and  we 
have 

mA  +  fiB  +  (7=0 

for  the  equation  of  condition  of  parallelism  of  a  line  and  plane. 

If  the  line  coincides  with  the  plane,  it  will  be  indetermi- 
nate and  the  numerator  will  also  be  zero. 

213.  Conditions  existing  between  a  Plane  and  a 
right  line  perpendicular  to  the  Plane. — Let  ABC\>q  Sk 
plane  and  DP  a  line  per- 
pendicular to  it.  Every 
plane  passed  through  DP 
will  be  perpendicular  to 
the  given  plane  since  it 
contains  a  line  perpen- 
dicular to  that  plane. 
Let  any  plane  be  passed 
through  it  and  revolved 
about  the  line  until  it 
is  perpendicular  to  the 
plane  zx;  the  trace  T8 
of  the  plane  in  this  posi- 
tion will  be  the  projec- 
tion of  the  line  on  the  plane  zx.  But  since  the  projecting 
plane  will  be  perpendicular  both  to  the  given  plane  and  the 
plane  zx,  its  trace  TS,  will  be  perpendicular  to  the  trace  A  B 
of  the  given  plane,  {Elementary  Geometry).  Similarly,  VQ 
will  be  perpendicular  to  CB,  and  UR  to  A  C.  Hence, — If  a 
line  be  perpendicular  to  a  plane,  its  projections  wiU  he  perpendicu- 
lar respectively  to  its  traces. 


Fig.  160. 


Let  Ax+By-hCz  +  D  =  0 

be  the  plane,  and 

x=  mz  +  a,         y  =  nz  +  b, 


(1) 
(2) 


the  line  perpendicular  to  the  plane.     Making  y  =  0  in  the 
equation  of  the  plane,  we  have 


Ax  +  Cz  +  D  =  0, 


164  LOCI  IN  SPACE.  [214,  215. 

CD 

for  the  trace  of  tlie  plane  on  x.c.  But  the  condition  of  per- 
pendicularity requires,  (Art.  44), 

l  +  »(-^)  =  0; 

.-.  A=mG.  (3) 

Similarly,  we  would  find 

B=^na  (4) 

214.  Equations  to  a  Line  perpendicular  to  a  given 
Plane. — Let  Ax  +  By  +  Cz  +  D  —  d  be  the  plane.  The 
equations  of  the  line  will  be  of  the  form 

X  =  mz  +  a,         y  =  nz  +  b, 

in  which  substitute  the  values  of  m  and  n  from  the  preced- 
ing Article,  and  we  have 

A  B       ^  ,n\ 

x=^z  +  a,  y  =  -^8  +  o;  (5) 

which  are  the  required  equations.  Since  the  arbitrary  con- 
stants a  and  b  remain,  there  may  be  an  infinite  number  of 
lines  perpendicular  to  a  given  plane. 

215.  Equations  to  a  Line  perpendicular  to  a  given 
Plane  and  passing  through  a  given  Point. — Let  {x,  y',  z') 
be  the  point.     The  equations  will  be  of  the  form 

x  =  -^z  +  a,  y=-^z  +  b; 

and  since  the  line  contains  the  given  point,  we  have 

x'=  -^z  +  a;       y-Q^  +o; 


216, 217.]  LINE  AND  PLANE.  165 

and  we  find 

X-X   =:^{Z-Z),  y-y  =  —{z-z')',  (6) 

which  are  the  required  equations. 

To  find  where  this  perpendicular  pierces  the  plane,  elimi- 
nate X  and  y  between  equations  (6)  and  (1)  and  find  z.  Simi- 
larly, find  X  and  y.     The  length  of  the  perpendicular  will  be 


^  A,v'  +  By'  ^Cz'  +  D 
^{A'  +  B^-V  C-)    ' 


^{A'  +  B'+  C-) 

216.  Equations  to  a  Plane  p^rssing  through  a  point  and  per- 
pendicular to  a  given  line. — Let  the  point  be  (x', y', z), and  the 
line 

x=^7nz  +  a,         y  =  nz  +  b. 

The  equation  of  the  plane  will  be  of  the  form 
Ax  +  By+Cz  +  I)  =  0, 
and  for  the  point,  we  will  have 

Ax'  +  By'  +Cz'  +  I)  =  0. 
Subtracting,  we  have 

A{x-x')  +B{y  -y')  +  C{z-z')  =  0, 
But,  from  Article  213,  we  have 

A  =  mC,        B  =  nC; 
which  substituted  above  gives 

m{x—  x)  +  n{y  —  y')  +  z  —  z'  =  0, 
which  is  the  required  equation. 

217.  To  find  the  Angle  between  a  Line  and  Plane. 
It  will  be  the  angle  between  the  line  and  its  projection  on 
the  plane,  and  this  equals  the  complement  of  the  angle  be- 
tween the  line  and  a  perpendicular  to  the  plane.     From  the 


166  LOCI  IN  SPACE.  [218, 219. 

origin  draw  a  line  parallel  to  tlie  given  one,  and  let  its  equa- 
tions be 

X  =  mz,        y  =  nz. 

Draw  another   line   from  the  origin,  perpendicular  to  the 
plane  Ax  +  By  +  Cz  +  D  —  0,  and  let  its  equations  be 

X  =  m'z,        y  =■  n'z, 

then,  Article  213,  equations  (3)  and  (4), 

,      A  ,      B 

Let  Cbe  the  required  angle,  then  from  equation  (6)  Article 

199,  we  have 

_    .     „_  mA  +  nB  +  C 

cos  rp-SmU-  ^^(1  +  ^2  ^  ^.2)  (^2  ^  ^^  ^. 

If  the  line  is  parallel  to  the  plane,  we  have  sin  U=0, 

therefore, 

mA  +  oiB  +  C=0, 

as  before  found,  (Art.  212). 

Transformation  of  Coordinates. 

218.  The  sum  of  the  projections  of  any  number  of  hroJcen 
lines  joining  two  points  upon  the  right  line  joining  those  points, 
equMs  the  length  of  the  right  line. 

If  the  projections  all  fall  between  the  points,  the  propo- 
sition is  evidently  true.  If  any  of  the  projections  fall  upon 
the  prolongation  of  the  line,  there  will  be  positive  and  neg- 
ative projections  (the  signs  being  determined  by  the  sign  of 
the  cosine  of  the  inclination),  the  algebraic  sum  of  which 
will  equal  the  length  of  the  line. 

219.  To  transform  from  a  given  rectangular  system  to  a  sys- 
tem of  oblique  planes. 

Let  X,  y,  z  be  the  rectangular  axes,  x',  y',  z,  the  oblique 
axes ;  ^,  X'\  X'",  the  angles  between  x  and  x\  y',  z\  respec- 
tively ;  T',  T",  Y'",  between  y  and  the  same  axes ;  Z',  Z'\ 
Z  ",  between  z  and  the  same  axes. 

The  projection  of  the  co'Tdinates  x ,  y',  z'  of  any  point 
apon  the  axis  of  x,  will  equal  the  x-absdssa  of  the  point 


220.] 


TPiAXSFORMATlON  OF  COORDmATES. 


167 


(2) 


(3) 


For,  tlie  tliree  oblique  coordinates  projected  on  the  radius- 
vector  of  the  point,  will  equal  the  radius-vector,  and  the 
projection  of  the  radius-vector  on  the  axis  of  x  will  equal 
the  x-abscissa  of  the  point.  Similarly  for  the  other  axes. 
Hence  we  have 

X  —  X  cos  X'  +  y'  cos  X"  +  z'  cos  X'  \ 

y  =  x'  cos  Y'  +  ])  cos   Y  '  +  z  cos  X'",  (1) 

z  —x  COS  Z'  +  y'  cos  Z"  -\-  z  cos  Z' '. 

If  the  coordinates  of  the  new  origin  be  a,  h,  c,  we  would 

have 

X  —  a  +  x'  cos  X'  -T-  y'  cos  X"  +  z'  cos  X'", 

y  =  b  +  x'  cos  Y'  +  y'  cos  Y"  +  z  cos  Y"\ 
z  —  c  ^  x  COS  Z'  -r  y'  COS  -^"  +  z  cos  Z". 
The  angles  are  subject  to  the  condition 

cos'X '  +  cos-  Y '  +  cos-  Z'  =1, 
cos^X"  +  cos^  Y'  +  cos-^"  =  1, 
cos-X'"  +  cos^r"'+  cos-Z'  '=  1. 

If  the  angles  between  the  oblique  axes  are  sought,  we 
have  (Ari  199,  Eq.  (5)), 

cos  {x'y')  =  cos  X'  cos  X"  +  cos  Y'  cos  Y"  -!-  cos  Z'  cos  Z'\ 
cos  (?/V)  =  cosX'  cosX"'+  cos  7"  cos  F"'+  cosZ"  cosZ"',{4i) 
cos  (s'a/)^  cosX'  co8X"'+  cos  Y'  cos  F"+  cos^'  cos^'". 

Equations  (3)  and  (4)  will  make  known  the  angles  between 
the  oblique  and  the  corresponding  rectangular  axis. 

220.  Polar  Coordinates. — Let  0  be  the  pole ;  P  any 
point ;  p  —  OP  =  the 
radius  -  vector  of  the 
point;  OX  the  initial 
line  ;  XOZthe  initial 
plane;  OD  the  pro- 
jection of  the  radius- 
vector  ;  6  =  BOX  = 
one  variable  angle, — 
corresponding  to  the 
azimuth  angle  in  as-  fig.  lei. 


168  LOCI  IN  SPACE.  [231. 

tronomy ;  (p  =  POD,  the  other  variable  angle, — correspond- 
ing to  the  angle  of  elevation  in  geodesy,  or  of  declination 
in  astronomy.     Draw  DF  perpendicular  to  OX,  and  we  have 

OF  —  x—p cos  cp cos  B, 
DF  =  y  —  p  cos  cp  sin  B, 
PD  =  2=  p sin  q) ; 

by  means  of  which,  rectangular  coordinates  may  be  changed 
to  polar. 

221.  To  find  formulas  for  passing  from  a  polar  system  to  a 
rectangular  one. 

For  this  we  find  ^,  B,  and  p  from  the  preceding  equations. 
We  have 

a?  +  y^  -^^  z^  =  p^ 


sm«2>  =  -;  .'.  cos  ©  =  r  1 ^. 

P  P" 

co8^  = ;  or  sin  ^= — - — : 

pcoscp  pcosq) 

from  which  B  and  (p  may  be  found  in  terms  of  x,  y,  and  s. 


EXAMPLES. 

1.  What  is  the  distance  between  two  points  whose  coSrdinates  are 

«'  =  —  2,  y  =  1,  s'  =  0 ;  and  x"  =  0,  y"  =  —  5,  z"  =  4  ? 

2.  The  equations  of  the  projections  of  a  straight  line  on  the  coordi- 
nate planes  zx,  yz,  are 

1 

x  =  z  +  l,  y  =  -z-2, 

required  its  equation  on  the  plane  px. 

Ans.  2y  =  a;  —  5. 
3.  Required  the  equations  of  the  three  projections  of  a  straight  line 
which  passes  through  the  two  points  whose  coordinates  are 

x'  =  2,  y'  =  1,  z'  =  0,  and  x"  =  -  3,  y"  =  0,  z"  =  -  1. 

Ans.  x  =  5z  +  2,  y  =  z+  1,  5y  —  x+d. 


331.]  EXAMPLES.  169 

4.  Required  the  angle  included  between  two  lines  whose  equations 

are 

x—Zz  +  5)  X  =  Z  +  1  ) 

(^^  ,   Q  ^  of  the  1st,  and         „         [  of  the  2d. 

Ans.   14°  58'. 

5.  Required  the  angles  which  a  straight  line  makes  with  the  coordi- 
nate axes,  its  equations  being 

a;  =  —  23  +  1,  y=  z  +  3. 

144°  44'  with  X, 


Ans.    ■{    Go"  54'  with  Y, 
65'  54'  with  Z. 


6.  Having  given  the  equations  of  two  straight  lines, 


=  23+1)  X=2+5) 

n    ,   o  i  01  the  1st,  and  .        ,j.  c  of  tlie  2d, 

=  23  +  2  )  '  y  =  43  +  /i    )  ' 


a;=  23  + 

y 


required  the  value  of  /3'  so  that  the  lines  shall  intersect  each  other,  and 
to  find  the  coordinates  of  the  point  of  intersection. 

Am.  ft'  =  -  6,  a;'  =  9,  t/'  =  10,  z  =  4. 

7.  To  find  the  equations  of  a  line  that  shall  pass  through  a  point,  of 
which  the  coordinates  are  x'  =  —  2,  y'  =  3,  z'  =  5,  and  be  perpendicular 
to  the  plane,  of  which  the  equation  is 

2x  +  8y  -  z  -  4  =  0. 

x=  —  2z  +  8, 


^««-    )  2/= -83  +  43. 

8.  To  find  the  equation  of  a  plane  which  shall  pass  through  the  three 
points,  whose  coordinates  are 

x'  =  1,     y'  =z  —  2,     z'  =  2;        x"  =  0,     y"  =  4,     s"  =  -  5 ; 

x'"  =  -  3,     y'"  =  1,     3'"  =  0. 

Ans.  9a;  +  19y  +  153  —  1  =  0. 

9.  To  find  the  equations  of  the  intersection  of  two  planes,  of  which 
the  equations  are 

3a;  +  8y  —  103  +  6  =  0,  of  the  1st, 

and  4a;  —  8y  +  3     +  1  =  0,  of  the  2d. 

j    7a;  —  93  +  7  =  0, 
^"«-    \  56y  -433  +  21  =  0. 


170  LOCI  IN  SPACE.  [231. 

10.  To  find  the  traces  of  a  plane  whose  equation  is 

x-Qy  +  llz—  12=0. 

11.  To  find  the  length  of  a  line  drawn  from  a  point  whose  coordi- 
nates are  x'=  2,  y'  =  —  3,  2'=  0,  and  perpendicular  to  a  plane  whose  equa- 
tion is 

8a5+9y  —  2  +  2  =  0. 

Ans.   ■  , :. 

12.  Find  the  equation  of  a  plane  passing  through  the  point  x  =  1,  y' 
=  —  3,  0'  =  4  and  perpendicular  to  the  line  a;  =  82  —  4,  y  =  —  22  +  5. 

13.  Required  the  angle  between  the  two  planes 

5x  —  7y  +  32  + 1  =  0,  and  2x  +  y—  3z  =  0. 

Ans.  lOO""  8'. 

14.  Required  the  angle  which  the  plane  5x  —  7y  +  B2  +  1  =  0,  makes 
with  the  coordinate  planes. 

(    70°  46' with  Xr. 

Ans.    }  140°  12'  with  ZX. 

i    56°  43'  with  YZ. 

15.  Find  the  equation  of  a  right  line  Avhich  passes  through  the  point 
(a,  b,  c)  and  is  perpendicular  to  each  of  two  right  lines  whose  direction- 
cosines  are  Z,  m,  n ;  I',  m\  n. 

X —  a  y  —  b  z—  c 


Ans. 


nl  —  n'l      Im'  —  I  'm 


16.  Find  the  equations  of  a  line  in  space  in  terms  of  the  direction- 
cosines  of  its  angles  with  the  three  axes. 

17.  Find  the  equation  to  a  plane  passing  through  the  points  (2,  3,  4), 
(3,  4,  5),  and  perpendicular  to  the  plane  x  +  4y  +  22  =  1. 


CHAPTER  VII. 


CURVED     SURFACES. 


222.  Definitions. — A  line  may  be  generated  by  the 
movement  of  a  point ;  a  surface  by  the  movement  of  a  line  ; 
and  a  volume  by  the  movement  of  a  surface. 

TJi£  generatrix  is  the  moving  element,  whether  it  be  a 
point,  line,  or  surface. 

The  directrix  is  a  fixed  element,  about  which  the  genera- 
trix moves. 

A  surface  of  revolution  is  a  surface  generated  by  the 
revolution  of  a  line  about  an  axis. 

A  curved  surface  is  one  from  which  a  curve  may  be  cut 
by  a  plane. 

A  single-  curved  surface  or  surface  of  single  curvature  is  one 
which  may  be  generated  by  a  right  line  having  its  consecutive 
positions  in  one  plane  ;  as  a  cylinder,  or  a  cone. 

A  surface  of  double  curvature  is  one  which  can  be  gener- 
ated by  a  curve  only ;  or  one  from  which  a  right  line  cannot 
be  cut ;  as  a  sphere,  ellipsoid,  paraboloid,  etc. 

A  warped  surface  is  one  which  may  be  generated  by  a 
right  line  no  two  consecutive  positions  of  which  are  in  one 
plane  ;  as  the  hyperbolic  paraboloid,  conoids,  etc. 

An  Jiyperholic  paraboloid  is  a  surface  from  which  parab- 
olas may  be  cut  by  a  certain  system  of  parallel  planes, 
and  hyperbolas  by  another  system  of  parallel  planes,  (Art. 
236). 

A  canmd  may  be  generated  by  a  line  moving  around  a 
curve,  while  some  point  of  the  line  moves  to  and  fro  along  a  right 
line,  the  generatrix  remaining  parallel  to  a  plane.     The  plane 

171 


172  CURVED  SURFACES.  [323,234 

is  called  the  plane-directer,  and  is  generally  perpendicular  to 
the  right-line  directer. 

A  line  of  single  curvature  is  one  which  changes  its  direction 
at  every  point,  and  all  of  whose  points  are  in  one  plane ;  as 
a  circle,  ellipse,  spiral,  etc. 

A  line  of  double  curvature  is  one  which  changes  its  direc- 
tion at  every  point,  and  all  of  whose  points  cannot  lie  in 
one  plane ;  as  the  thread  of  a  screw ;  a  thread  wound  spi- 
rally around  a  cone,  or  sphere,  etc. 

Of  Cylindrical  Surfaces. 

223.  A  Cylinder  may  be  generated  by  a  right  line 
moving  around  a  fixed  curve  and  remaining  parallel  to  a  fixed 
line.  The  fixed  curve  is  the  directrix,  and  the  moving  line, 
the  generatrix. 

If  the  generatrix  is  perpendicular  to  the  plane  of  the  di- 
rectrix, the  cylinder  is  called  right.  An  oblique  cylinder  is 
one  in  which  the  generatrix  is  inclined  to  the  plane  of  the 
directrix.  A  circular  cylinder  is  one  in  which  any  right  section 
is  a  circle.  An  elliptic  cylinder  is  one  in  which  any  right  section 
is  an  ellipse.  Generally  a  cylinder  takes  its  name  from  the 
character  of  its  right  sections,  or  by  calling  it  an  oblique  cyl- 
inder with  a  given  base,  thus  we  may  say  an  oblique  cylinder 
with  a  circular  base,  or  an  oblique  cylinder  with  an  ellipti- 
cal base,  etc. 

224.  Equations  of  a  Right  Cylinder.— 1°.  Let  the  base 
be  a  circle.  Take  the  origin  at  the  centre 
of  the  circle,  the  plane  xy  in  the  plane  of 
the  circle,  and  the  axis  s  perpendicular  to 
that  plane;  then  will  the  generatrix  be 
parallel  to  the  axis  of  z.  Let  F  be  any 
point  in  the  surface,  and  CEP  any  section 

^  parallel  to  the  base.     Let  fall  the  perpen- 

^''-  ^^-  dicular  PD  upon  CE,  the  radius  CE  being 

parallel  to  the  axis  of  « ;  then  will  the  coordinates  of  P  be 
2=  CO,  x=  CD,  y  =  DP. 

Let  r  =  CE  =  CP=  the  radius  of  the  base  ;  then 

r'=CI)'  +  I)P^; 

or  r^  =  a^  +  y\ 


^:!^ 


225.]  OF  CYLINDRICAL  SUliFACES.  173 

and  z  indeterminate;  hence  the  equation  of  the  surface  will  be 

c(?  +  y-  +  O.z-  —  r-. 

2°.  Let  the  hose  he  an  ellipse. — Proceeding  as  before,  we 
find 

b'x'-hay-  +  0.s'  =  aW, 

for  the  equation  of  the  surface. 

3^.  Let  the  base  he  a  parabola. — Take  the  origin  at  the  ver- 
tex, the  plane  of  the  curve  being  in  the  plane  xy  ;  then  we 
find 

y^  +  0.s=  2px. 

225.  Equation  of  an  Oblique  Cylinder. — Let  the  di- 
rectrix be  in  the  plane  xy  and  represented  by 

F(x,y)=0,  (1) 

which  is  read,  fun/;tion  xy  equal  zero.  It  implies  that  x  and  y 
are  dependent  upon  each  other,  and  may  be  made  to  repre- 
sent the  coordinates  of  any  plane  curve.     For  instance, 

for  the  circle,         F  {x,  y)  =x^ -\-  y'^  —  r^  =  0\ 
for  the  ellipse,       F  {x,  y)  =  6 V  +  a^  —  a%^  =  0 ; 
for  the  parabola,    F  {x,  y)  =y^  —  ^px  =  0  ; 
for  the  right  line,  F  (x,  y)  =  y  —  mx  —b=0. 

For  the  oblique  cylinder,  the  equation  of  the  generatrix 
will  be,  (Art.  195), 

x=  mz  +  a,        y  =n2  +b,  (2) 

from  which  we  have 

a  =  X  —  mz,        b  =  y  —  nz  ;  (3) 

in  which  a  and  b  are  the  coordinates  of  the  point  where  the 
generatrix  pierces  the  plane  xy.  But  to  generate  the  re- 
quired cylinder,  the  point  (a,  b)  must  move  around  on  the 
directrix,  and  its  coordinates  become  the  coordinates  of  the 
directrix  ;  hence  they  take  the  place  of  x  and  y  in  the  equa- 


174  CURVED  SURFACES.  [225. 

tion  of  the  directrix,  and  the  furictional  equation  of  the  surface 

becomes 

F  («,  h)  =  F{x-m2,y-  m)  =  0.  (4) 

In  this  equation,  x,  y,  s,  are  the  coordinates  of  the  surface 
and  a,  b,  the  values  of  x,  y,  when  z  =  0.  The  particular 
value  of  F  (a,  b)  will  be  determined  from  the  character  of 
the  base. 

Applications. — 1°.  Let  the  base  be  a  circle.  Then  will  the 
equation  of  the  base  be 

Fia,b)=a'+b' -7^  =  0. 

Substituting  in  this  equation  the  values  of  a  and  b  from 
equations  (3),  give 

(x  —  mzy  +  {y  —  my  =  r^ ; 

which  is  the  equation  of  an  oblique  cylinder  having  a  circular 
base. 

If  the  generatrix  be  parallel  to  the  axis  of  s,  then  m  =  0, 
and  n  =  0,  and  we  have 

as  previously  found  for  the  equation  of  a  right  circular  cyl- 
inder. 

2°.  Let  the  base  be  a  parabola.  Take  the  origin  at  the  ver- 
tex, and  we  have 

F{a,b)  =  a''-2pb=.0; 

and  the  equation  of  the  surface  becomes 

{x  —  mzf  —  2p  {y  —  nz)  =  0  ', 

which  is  the  equation  of  the  surface  of  an  oblique  cylinder 
having  a  parabola  for  its  base. 

3°.  Let  the  base  be  an  ellipse.     We  will  have 

b-  (x  —  razf  +  a^  (y  —  nzf  —  c^l? 
for  the  required  equation. 

4°.  Let  the  directrix  be  a  right  line.  We  will  have,  (Art 
28), 

F  (a,b)  =  b  —  mia  —  Jj  =  0 ; 


226,  227.] 


OF  COXICAL  SURFACES. 


175 


in  wliicli  mi  and  hi  are  used  so  as  to  distinguisli  them  from 
the  letters  in  the  preceding  equations. 

Substituting  the  values  of  a  and  h  from  equation  (3), 
we  have, 

(y  -lu)  -  mi{x  -  mz)  -  &,  =  0, 

which  is  the  equation  of  a  plane,  (Art.  202). 


Of  Conical  Surfaces. 

226.  A  Conical  Surface  may  be  generated  by  a  right 
line  constantly  passing  through  a  fixed  point  and  moving 
around  a  fixed  curve. 

The  moving  line  is  the  generatrix,  the  fixed  curve  the  di- 
rectrix, and  the  fixed  point  the  vertex  (or  apex)  of  the  cone. 

The  line  will  generate  two  parts  of  the  surface,  one  part 
being  on  one  side  of  the  fixed  point,  and  the  other  part  on 
the  opposite  side  ;  each  of  these  parts  is  called  a  naj)pe  of 
the  cone. 

A  cone  is  described  from  the  character  of  its  base  and 
inclination  of  its  axis  ;  thus,  we  may  have  a  right  or  oblique 
cone  with  any  given  base.  A  rigJit  cone  is  one  in  which  the 
line  joining  the  apex  with  the  centre  of  the  base  is  perpen- 
dicular to  the  plane  of  the  base.  If  the  base  has  not  a 
centre,  the  cone  cannot  properly  be  called   right. 

227.  Equations  to  the  surface  of  a  Right  Cone.— 
1°.  Let  the  hose  he  a  circle.  Take  the  ori- 
gin at  the  vertex,  and  the  plane  xij  par- 
allel to  the  plane  of  the  circle,  then  will 
the  axis  of  z  coincide  with  the  axis  of 
the  cone.  Let  P  be  any  point  in  the 
surface,  and  pass  a  plane  through  it 
parallel  to  the  plane  xy  ;  its  section  will 
be  a  circle.  The  coiirdinates  of  P  will 
be 

x=CD,        y=DP,        z=CO. 

Let  h  =  AO=:  the  altitude  of  the  cone, 
AB  —  r  =  the  radius  of  the  base,  and 
V  =  AOB=  the  angle  between  an  ele-  fig.  lea. 


176  CURVED  SURFACES.  [238. 

meut  dJid  the  axis ;  then 

CP  ^  CE  =  ziojiv. 

But  CD^  +  DP^  =  CP'''y 

or  3(^  +  y^  =  z''  tan^  v, 

which  is  the  equation  of  a  right  cone  having  a  circular  b^se. 
2°.  Let  the  base  he  an  ellipse.     The  horizontal  section  CE 
will  be  similar  to  the  base.     Let  AB  =  a  be  the  semi-major 
axis,  h  the  semi-minor  axis,  and  we  have 

OA  :  OC  ::  AB  :  CE; 
or  h  :  B  ::  a  :  CE=  j-^=  a'  (say). 

Similarly,  CG  =-TB=b'  (say). 

For  the  point  P  of  the  ellipse,  we  have 
aY  +  b''x'  =  a"'b'K 
Substituting  the  values  of  a'  and  b',  we  have 

a^b^ 

which  is  the  equation  of  a  right  cone  having  an  elliptical 
base. 

228.  Equation  to  the  surface  of  an  Oblique  Cone. — 

Take  the  plane  xy  in  the  plane  of  the  directrix.  The  equa- 
tion of  the  directrix  will  be  F  {x,  y)  =  0.  The  equations  of 
the  right  line  which  forms  the  generatrix  will  be  of  the  form 

x  =  mz  +  a^        y  =  nz  +  b;  (1) 

but  since  this  line  must  pass  through  a  fixed  point  {x,  y',z'), 
its  equations  become 

x  —  x'  =  m{z—  z'),       y  —  y'  =  n{z  —  z'),       (2) 

or  x  =  mz  +  {x'  —  mz'),       y  =  m  +  (y'  —nz'),      (3) 


228.]  OF  CONICAL  SURFACES.  177 

in  wliicli  the  absolute  terms  {x  —  mz'),  (y'  —  nz'),  are  the 
coordinates  of  the  points  where  the  generatrix  pierces  the 
plane  xy ;  their  values  in  terms  of  the  general  variables  are 

[x  —  mz')  —X—  mz  ) 

ly'  -  nz')    ^y  -nz   ]'  (  ^ 

But  m  and  n  will  constantly  vary,  and  should  be  ex- 
pressed in  terms  of  the  general  variables.  Substituting 
their  values  from  equations  (2),  and  representing  the  left 
members  of  (4)  respectively  by  31^  and  Ny,  gives 


Mx  =  {x'  —  mz')  = J- 


(5) 


z  —  z 
and  the  general  equation  of  the  surface  becomes 

F(M„N,)  =  Fi^^^i^)=^.         (6) 

Applications. — 1°.  Let  the,  hose  he  an  ellipse.     The  coordi- 
nates of  the  base  being  M^.  and  N^, ,  its  equation  will  be 

aW/  +  b'3I^'  =  a'V ; 

in  which  substituting  the  values  of  M^  and  Ny  from  equation 
(5)  gives 

for  the  general  equation  of  an  oblique  cone  having  an  ellip- 
tical base. 

2°.  Let  the  cone  be  right.    The  vertex  will  be  vertically 
over  the  origin,  and  we  have 

x'  =0,        y'  =  0,        s'  =  h, 
12 


178  CURVED  8UBFACES. 

and  the  preceding  equation  becomes 


[239. 


or 


(8) 


which  will  become  the  same  as  the  last  equation  of  the  pre- 
ceding Article  if  the  origin  be  transferred  to  the  vertex. 

3°.  Let  the,  base  he  a  cirde  and  the  coiie  right.     Then  will  a 
=  6  in  the  preceding  equation,  and  we  have 

h\x'+y'')  =  {2  —  hya^  (9) 

for  the  equation  of  the  surface. 

4°.  Let  the  directrix  he  a  right  line.     The  equation  of  the 
directrix  will  be 

F{M^  ,Ny)  =  Ny-rrHM^-  h,=0. 

Substitute  the  values  of  N^,  and  M^.  from  equation  (5)  and 
we  have 

y's  —  z'y  —  rrii  {x's  —  z'x)  =  hi{z  —  2'), 

which  is  the  equaticm  of  a  plane,  (Art.  202). 

Surfaces  of  Revolution. 

229.  The  Sphere  is  a  surface  every  point  of  which  is 
equally  distant  from  a  point  within 
called  the  centre.  It  may  be  generated 
by  the  revolution  of  a  semicircle  about 
a  diameter.  Let  r  be  the  radius  of 
the  sphere.  The  origin  being  at  the 
centre,  the  distance  of  any  point  P  from 
the  centre  will  be,  (Art.  193),\/«M^yHf^ 
Fig.  164.  "but  this  cquals  the  radius ;   hence  the 

required  equation  is 

a?  +y^  +  z^  =  7^. 

If  the  coordinates  of  the  centre  be  a,  h,  c,  the  equation 


will  be 


{a  -  xf  +{h  -yf  +  {0-  zf  =7^. 


230,  231.J 


OF  ELLIPSOIDS. 


179 


230.  General  Equation. — When  a  surface  is  generated  by 
the  revolution  of  a  plane  curve, 
let  the  axis  of  s  coincide  with  the 
axis  of  revolution,  and  the  plane 
of  the  generatrix  be  in  the  plane 
zx.  Every  point  of  the  curve 
will  describe  a  circle  having  its 
centre  on  the  axis  of  s.  The 
length  of  the  radius  of  this  circle 
will  generally  depend  upon  the 
ordinate  z,  which  may  be  expressed  in  the  form 

p=/C^),  (1) 

from  which  p  may  be  found  when  the  equation  of  the  gen- 
eratrix is  known.     But  we  also  know,  (Art.  57),  that 


Fig.  165. 


p  =  Vx'  +  if ;  (2) 

•••  x^  +  f={f{^)\  (3) 

which  may  be  called  HdlQ  functional  equation  of  any  surface  of 
revolution. 

To  apply  this  method  to  the  sphere,  the  equation  of  the 
generatrix  will  be 

^2  ^  p2  _  ^  . 

.-.  f^={f{^)f  =  r'-z\ 

and  equation  (3)  becomes 

x"  +  y''+  s-  =  r\  (4) 

as  before  found. 

Of  Ellipsoids. 
231.  The  Prolate  Ellipsoid  may  be  generated  by  the 
revolution  of  a  semi-ellipse  about  its 
major  axis.  Let  the  semi-major  axis 
of  the  generating  curve  be  6,  the  semi- 
minor,  a.  Let  P  be  any  point  of  the  gen- 
erating curve,  p,  the  radius  of  the  circle 
which  it  describes ;  then  will  the  equa- 
tion of  the  generatrix  be 


180  CURVED  SURFACES.  [233-234 

Hence  equation  (3)   of  tlie  preceding  Article,  gives 


a? 


+C+S-1, 


(5) 


whiclL  is  the  equation  of  the  surface  of  the  prolate  ellipsoid. 

232.  The  Oblate  Ellipsoid  may  be  generated  by  the 
revolution  of  an  ellipse  about  its  minor  axis.  Let  the  axis 
of  3  be  the  axis  of  revolution.  Then  proceeding  as  before, 
we  have 


+  ^+;,2  =  l- 


(6) 


233.  The  Ellipsoid  of  the  most  general  character  is  a 
surface  such  that  all  sections  of  it  by  a  plane  are  ellipses. 
It  may  be  conceived  to  be  generated  by  an  ellipse  revolving 
about  one  axis  while  the  other  axis  increases  or  diminishes 
in  such  a  way  as  to  generate  an  ellipse  during  the  revolution, 
and  at  the  same  time  ihe  generating  curve  be  an  ellipse  in 
all  positions. 

Let  the  semi  axes  of  the  ellipsoid  be  a,h,G',  then  the 
equation  of  the  surface  will  be 


«2  +  j2-r^  --•- 


(7) 


Of  Paraboloids, 

234.  The  Paraboloid  of  Revolution  may  be  generated 
by  the  revolution  of  the  parabola  about 
its  axis.  To  find  the  equation  of  the 
surface,  let  the  axis  of  x  be  the  axis  of 
revolution,  and  P  any  point  in  the  sur- 
face. If  the  plane  of  the  generating 
curve  be  in  the  plane  zx,  its  equation 
will  be  z^  =  2px.  Let  PD  be  perpen- 
dicular to  EB ;  then 


Fie.  167. 


EB=PB=^^px, 


235.] 
But 

or 


OF  PARABOLOIDS. 

2p^2p     1  ~    ' 


181 


which  is  the  equation  of  the  surface  of  a  paraboloid  of  revo- 
lution when  revolved  about  the  axis  of  x. 

235.  Elliptical  Paraboloid. — Let  the  generatrix  be  a 
parabola  and  the  directrix  an  ellipse,  the  parameter  of  the 
parabola  being  supposed  to  vary  constantly,  so  that  the  gen- 
erating arc  may  constantly  pass  through  a  point  on  the 
ellipse.     The  surface  thus  generated  is  an  eUiptical paraboloid. 

Take  the  centre  of 
the  ellipse  on  the  axis  of 
X,  and  its  plane  parallel 
to  the  plane  zy.  Let  P 
be  any  point  on  the  sur- 
face whose  coordinates 
are  x,  y,  z,  the  origin  be- 
ing at  the  vertex  of  the 
parabola,  and  2p  the  pa- 
rameter of  the  parabola 
POG.     Draw  PD  perpendicular  to  CD,  then  we  have 

PC^  =  2px, 
or  ^  -{■  y^  —  2px, 


Fig.  168. 


(1) 


which  would  be  the  required  equation,  provided  that  p  varied 
according  to  the  required  law.  To  find  p  in  terms  of  general 
variables:  Let  bi=AC',  ai  =  B'C',  h=OC',  G  the  point 
where  the  arc  OP  intersects  the  ellipse,  and  whose  coordi- 
nates are  x^,  yi,  Zi,  and  2/),  the  parameter  of  the  parabola 
B'OB'.    Then 

a^'.=  2pih;     GG''=2ph  =  y,-  +  zi';  (2) 

2  «.  2     ,     <,  2 


...  2p  =  ^'  Y'    =^^-^^^P^' 


(3) 


182  CURVED  SURFACES. 

The  equation  of  tlie  ellipse  gives 


[23G. 


<hW  +  h'y,'  =  (h\'\  .'.  (h'  = 


,_  aiV+6,V 


^'l'^ 


which,  substituted  in  the  preceding  equatioif,  gives 


2p 


^1  +  y\ 


ttiW 


&iV 


2pA\ 


(4) 


(5) 


Since  this  equation  does  not  contain  x,  it  will  be  true  for 
all  corresponding  values  of  y  and  z.  Hence,  dropping  the 
subscripts  to  s  and  y  and  substituting  in  equation  (1),  gives 


aiV  +  6iV'  =  2i)i&i'ic, 


(6) 


for  the  required  equation.     Letting  2pibi^  =  dy,  and  dividing 
through  by  Oxhidi,  it  may  be  put  under  the  form 


(7) 


in  which  a^,  h\  d,  are  used  for  the  new  denominators  of  z^,  y\ 
X,  respectively. 

236.  The  Hyperbolic  Paraboloid  may  be  generated  by 
the  movement  of  a  parabola  whose 
parameter  so  varies  that  its  arc 
shall  follow  the  opposite  branches 
of  an  hyperbola,  the  vertex  re- 
maining at  the  same  point.  Let 
the  origin  be  at  the  vertex  of  the 
parabola,  the  plane  zy  be  parallel 
to  the  plane  of  the  hyperbola,  and 
the  axis  of  x  pass  through  the  cen- 
tre of  the  hyperbola.  If  the  equa- 
tion to  the  hyperbola  be  a^z^ — 
Wy\  =  ciihi,  we  would  find,  in  the 
^o- 169-  same  manner  as  shown  in  the  pre- 


237.]  OF  PARABOLOIDS.  183 

ceding  Article,  that  tlie  equation  of  the  surface  would  be 

a-     Ir      d        ' 

but  if  thfe  equation  of  the  hyperbola  be  a-^z-^  —  hiy^=  — 
ct^hi^  (which  is  the  conjugate  of  the  preceding  one),  then  we 
would  find 

a2  +  62      ^      ^ 

for  the  required  equation. 

237.  Problem. — Required  the  equation  of  the  surface  gen- 
erated by  a  light  line  moving  parallel  to  a  plane  and  along  any 
othei^  two  right  lines. 

Let  one  of  the   lines,  as  AB,  lie  in  the  plane  xy,   and 
the  other  in  the  plane  xz.     Take  the  origin  on  the  line  0  G. 
Let  the  equation  of  the  line  OC  be 
z  =  nx, 


of  AB,         y  =  mx  +  h, 

and  let  the  generating  line  CE 
be  constantly  parallel  to  the 
plane  zy,  then  will  its  projec- 
tion, EF,  on  the  plane  xy  be 
parallel  to  the  axis  of  y.  Let 
P  be  any  point  in  the  surface 
whose  coordinates  are  x  =  OF, 
y  =  FD,z  =  DP.     Then  will 


Fig.  170. 


and 


EF=7nx  +  &, 
CF=7ix. 


We  have  ED  .  DP  ::  EF  i  FG\ 

or  mx  +  b—  y  :  z  : :  mx  +  b  :  nx-, 

.'.  nm^  +  bnx  —  nxy  —  mzx  —bz  =  0, 


184 


CURVED  SURFACES. 


[238,  239. 


which  is  the  required  equation,  and  is  an  equation  of  the 
second  degree  between  three  variables.  The  character  of 
the  surface  is  not  readily  seen  from  this  equation,  but  by  a 
transformation  of  coordinates  it  may  be  reduced  to  the  form 

Mz^  -Ny^-  Qx  =  0;    . 

and  hence  it  is  an  hyperbolic  paraboloid,  (Art.  236).  It  is  a 
warped  surface,  (Art.  222).  The  same  surface  would  be  gen- 
erated by  the  line  OC  moving  on  OA  and  CE  in  such  a  man- 
ner as  to  divide  those  lines  proportionally. 


Of  Hyperholoids. 

238.  If  the  hyperbola  EA  revolves  about  its  conjugate 

axis  NO,  the  surface  gen- 

D       erated  will  be  continuous, 

and  is  called  an  hyper- 

c^  ,^  boloid  of  revolution  of 

one  nappe. 

The   equation    of   the 
hyperbola  is 


FiQ.  171. 


NE^  = 


a^z^  +  a^l? 


But 


NE^  =  ND=  NM^  +  ME"  =  x' +  y^; 


"    ^'+f  =  T^ 


or,  dividing  by  a\  we  have 


(1) 


which  is  the  equation  of  the  hyperboloid  of  revolution  of 
one  nappe. 

239.  If  the  hyperbola  revolves  about  the  axis  of  x,  two 
surfaces  will  be  generated,  one  by  the  right  branch  of  the 
curve,  and  the  other  by  the  left.     This  is  called  an  hyperbo- 


240,  241.1  OF  HYPEP^BOLOLDS.  185 

loid  of  two  nappes.     Its  equation  is 

^  _f_t^l,  (2) 

a-      6-      U"  ^  ^ 

In  these  equations  the  negative  signs  apply  to  those  axes 
which  do  not  intersect  the  surface. 

240.  Elliptical  Hyperboloid. — If  the  arc  of  the  hyper- 
bola be  made  to  follow  an  elliptical  directrix,  the  surface 
generated  will  be  an  elliptical  hyperboloid.  Its  equation 
will  be  of  the  form 

%+i-i=^^  (3) 

a^      0-       cr 
when  the  surface  is  of  one  Kiappe  ;  and 

when  of  two  nappes. 

Observe  that  the  character  of  these  surfaces  may  be  de- 
termined by  finding  the  character  of  the  curve  of  intersec- 
tion of  the  coordinate  planes  with  the  respective  surfaces. 
Thus,  in  equation  (1)  if  2=0,  we  have  a?  +  'i^  —  a^,  which  is 
the  equation  of  a  circle.  If  x—  0,  we  have  Irif  —  err  =a-h-, 
which  is  the  curve  of  intersection  of  the  plane  yz  with  the 
surface,  and  is  an  hyperbola. 

In  equation  (4),  if  2  —  0,  we  have  the  equation  of  an  hyper- 
bola, which  is  the  intersection  of  the  plane  cry  with  the  sur- 
face. Similarly,  if  ?/  =  0,  we  have  the  equation  of  the  inter- 
section of  the  surface  by  the  plane  xz,  which  is  also  an 
hyperbola.  If  a:  =:  0,  the  curve  is  imaginary,  or  the  plane  yz 
does  not  cut  the  surface.  The  surface  represented  by  equa- 
tion (3)  is  cut  by  aU  the  coordinate  planes. 

241.  Trohlem..— To  ^find  the  surface  generated  hy  any  line 
revolving  about  another  line.  Take  the  axis  of  z  as  the  direc- 
trix, and  the  axis  of  x  perpendicular  both  to  the  directrix 


186 


CURVED  SURFACES. 


[241. 


and  tlie  generatrix,  and  passing  tlirougli  the  point  wliere  the 
plane  xz  cuts  the  given  line.  Then 
will  the  projection  of  the  generatrix 
on  the  plane  xz  be  parallel  to  the 
axis  of  z,  and  the  projection  on  zy 
will  pass  through' the  origin.  The 
line  will  make  a  constant  angle,  COZ, 
with  the  axis  of  z,  and  the  point  A 
will  describe  the  arc  of  a  circle.     Let 

tan  COZ=  I,  and  OA  =  d;  then  will  the  square  of  the  distance 

of  any  point  I)  from  the  axis  of  z  be 


Fig.  172. 


or,  Pz^  +  cP; 

hence  we  have  x^  +  y'^  =  l^z^  +  cP, 

or,  dividing  by  oP  it  may  be  written  in  the  form 

x^      y^      ^^  —  1 


(1) 


(2) 


which  is  the  equation  of  an  hyperboloid  of  revolution  of  one 

nappe,  (Art.  238).  Hence,  an 
hyperboloid  of  one  nappe  is  a 
warped  surface. 

[This  problem  may  be  solved  in  a 
more  general  way.  Let  the  equations 
of  the  generatrix  be 

x  =  mz  +  a,    p  =  nz  +  b      (3) 

in  which  a  and  &  are  the  cobrdinates 
of  the  points  where  the  generatrix 
pierces  the  plane  xy.  The  locus  of  this 
point  will  be  a  circle,  hence  we  have 

a*  +  62  =  r*  =  {x-mzy  +  (y  —  nz)-.    (4) 
Fig.  173. 

In  this  equation  m  and  n  are  variables,  but  they  must  be  subjected  to  the 
condition  that  the  angle  which  the  generatrix  makes  with  the  axis  of  z  is 
constant ;  hence,  (Art.  198), 


w*  +  w*  =  a  constant  =  d^,  (say), 


(5) 


242.]  OF  INTERSECTIONS.  187 

and  equation  (4)  becomes 

r-  =  a;2  +  y^  —  2{mz  .x  +  nz.y)  +  d^z^.  (6) 

Substituting  the  values  of  mz  and  nz  from  equations  (3),  we  find 

x~  +  y^  =  2{ax  +  by)  —  r-  +  d-z~  ;  (7) 

in  wMch  substitute  from  (3)  the  values 

ax  =  amz  +  a~,  hy  ■=  bnz  +  b-,  (8) 

and  we  find  x-  +  y-  =  2{am  +  bn)z  +  r^  -\-  d~z^.  (9) 

Here  are  four  arbitrary  constants  a,  b,  m,  n,  between  whicli  two  condi- 
tions have  been  fixed  ;  we  may  therefore  make  an  arbitrary  assumption  and 
leave  them  still  indeterminate.  Assuming  that  the  origin  of  coordinates  is 
fixed  by  the  condition 

am  +  6ft  =  0,  (10) 

we  finally  have  x^  +  y^  =  r^  +  d~z~,  (11) 

which  is  of  the  same  form  as  equation  (1). 
If  a  =  0,  we  have 

X'  +  y^  =  r^, 

which  is  the  equation  of  the  curve  of  intersection  of  the  plane  xy  with  the 
surface,  which  curve  is  a  circle. 
If  05  =  0,  we  have 

y^  —  d-z^  =  r-  ; 

which  is  the  equation  of  an  hyperbola,  and  is  the  equation  of  the  curve  of 
intersection  of  the  plane  yz  with  the  surface.  Similarly,  the  intersection  of 
the  plane  xz  with  the  surface  is  an  equal  hyperbola.  The  condition  of 
equation  (10)  places  the  origin  so  that  the  surface  will  be  symmetrical  in 
reference  to  the  axes.  All  horizontal  sections  are  circles,  and  that  part 
of  the  surface  who.se  section  is  least  is  called  the  gorge.  In  equation  (11) 
the  origin  is  at  the  centre  of  the  gorge.  In  equation  (9)  the  origin  may  be 
anywhere  on  the  axis  of  s. 

If  the  directrix  were  an  ellipse  the  surface  generated  by  the  line  would 
be  an  elliptical  hyperboloid  of  one  nappe;  and  similarly  for  other  direc- 
trices.] 

Of  Intersections. 

242.  Problem. — To  find  the  intersection  of  a  plane  and 
sphere. 

Tlie  equation  of  the  sphere  is 

a?^y'  +  ^  =  r\  (1) 


188  CURVED  SURFACES.  [24a 

Let  the  cutting  plane  be  parallel  to  the  plane  xy^  then,  in 
the  equation  of  the  plane 

Ax+By  +  Cz  +  D  =  0,  (2) 

will  A  and  B  be  zero,  and  we  will  have   , 

_  _D 


which,  substituted  in  the  equation  of  the  sphere,  gives 


i>2 


which  is  the  equation  of  a  circle.     It  is  real  when  -^g  is  less 

than  1^,  and  imaginary  if  it  is  greater  that  rl  It  will  be 
greatest  when  i)  =  0 ;  but  when  D  is  zero  the  plane  passes 
through  the  centre  of  the  sphere,  as  shown  by  equation  (2), 
since,  in  that  case,  equation  (2)  has  no  absolute  term,  and  the 
origin  is  at  the  centre. 

In  a  similar  manner  we  find  that  the  intersection  of  the 
sphere  by  any  plane  parallel  to  the  coordinate  planes  is  a 
circle. 

Generally,  substitute  the  value  of  z  from  equation  (2)  in 
equation  (1),  and  we  have 

{A^+  C^)  x''  +  {B'+C')f  +  2ABxy  +  2ABx+2BDy=  C'r^'-I^; 

which  is  the  projection  of  the  curve  on  the  plane  xy ;  and  is 
an  ellipse,  (Art.  178).  But  in  order  to  determine  the  charac- 
ter of  the  curve,  we  must  find  its  equation  in  its  own  plane, 
that  is  in  the  plane  the  equation  of  which  is  given  by  equa- 
tion (2).     This  process  will  be  explained  in  Article  249. 

243.  To  find  the  {intersection  of  a  j^ne  toith  an  hyperbolic 
paraJbohid. 

The  equation  of  the  hyperbolic  paraboloid  is,  (Art.  236), 

^-^!_^  =  0.  (1) 


243.]  OF  INTERSECTIONS.  189 

Let  the  plane  be  parallel  to  the  plane  zy,  then  will  its 
equation  be 

Ax  +  D  =  0,         y  and  z  indeterminate  ; 

D 

and  this  value  reduces  the  preceding  equation  to  the  follow- 
ing; 

2"       y-  ^  D 

which  is  the  equation  to  an  hyperbola,  (Art.  78),  hence,  AU 
sections  of  an  kyperbolic  paraboloid  made  by  a  plane  perpendicu- 
lar to  the  axis  of  the  pxirabolic  sectioiu,  are  hyperbolas. 

Let  the  plane  be  parallel  to  the  plane  xz,  then  will  the 
equation  of  the  plane  be 

By  +  D  =0 ,        X  and  z  indeterminate  ; 

D 

and  this  value  in  the  equation  of  the  surface  gives 

^_x      B' 
^-~d^VB" 

which  is  the  equation  of  a  parabola,  (Art.  88). 

Let  the  cutting  plane  be  parallel  to  xy,  and  the  equation 
of  the  plane  be 

z  =  g,        X  and  y  indeterminate. 

This  value  in  equation  (1)  gives 

¥~     d^  a'' 

T     (i^  .    X      C^ 

which  is  imaginary  if  -^  >  -j ,  and  a  real  parabola  if  -5  <  -^ . 


190  CTTBVED  SURFACES.  [244-247. 

244.  Surfaces  of  the  Second  Order  are  those  whose 
equations  are  of  the  second  degree.  It  will  be  observed 
that  all  the  curved  surfaces  discussed  in  this  chapter  are  of 
the  second  order. 

245.  The  intersection  of  a  surface  of  the  second  or- 
der by  a  Plane  is  a  Conic  Section. — For  the  equation  to 
the  curve  of  intersection  is  found  by  eliminating  one  of  the 
variables  between  the  equations  of  the  plane  and  surface. 
But  the  equation  of  the  surface  is  of  the  second  degree,  and 
of  the  plane,  of  the  first  degree  ;  hence,  according  to  the 
principles  of  algebra,  the  resulting  equation  will  be  of  the 
second  degree,  and  hence  will  be  the  equation  of  a  conic, 
(Art.  177).  The  curve  thus  found  is  the  projection  of  the 
required  curve  on  one  of  the  coordinate  planes,  but  by  trans- 
forming the  coordinates  so  as  to  represent  the  equation  of 
the  curve  in  its  own  plane,  the  degree  of  the  equation  is 
not  changed,  (Arts.  ld>la  and  187&). 

246.  Intersection  of  two  surfaces  of  the  second  or- 
der.— Proceeding  as  before  to  eliminate  the  variables  from 
the  equations  to  the  surfaces,  the  resulting  equations  will  be 
the  equations  of  the  projections  of  the  curves  of  intersec- 
tion on  the  respective  planes.  If  the  intersection  be  a  plane 
curve,  its  equation  may  be  found  in  its  own  plane.  Generally ^ 
however,  the  curve  of  intersection  of  surfaces  of  the  second 
order  will  be  a  curve  of  double  curvature,  (Art.  222),  in 
which  case  its  character  can  be  determined  only  by  consid- 
ering its  three  projections. 

247.  Intersection  of  a  sph&re  and  ellipsoid  of  revolution. 
Let  the  equations  be 

x^  +  y''-  +  z^  =  7^,        ax"  +  ay^  +  bz^  =  d. 
Eliminating  z  gives 

(a  -  hjx'  +  (a-  b)f  =  d  -  br\ 
which  is  the  equation  of  a  circle  whose  radius  is 


\/' 


d  -  br"^ 
a—  b 


248.]  OF  INTERSECTIONS.  191 

Eliminating  x  gives 


z  Va  —  h  =  \^aT^  —  df 

whicli  gives  a  point  on  the  axis  of  z.  If  the  intersection  of 
the  two  surfaces  be  not  a  point,  the  preceding  equation  will 
be  the  equation  of  a  line  and  may  be  written 


O.y  +  Va  —  b.z—  Var-  —  d, 

which  is  the  equation  to  the  trace  of  a  plane  on  yz  parallel 
to  the  plane  xy.  The  line  of  intersection  is  therefore  a  plane 
curve,  and  is  a  circle. 

248.  Intersection  of  a  sphere  and  hyperbolic  paraboloid. 
■  Let  the  equations  be 

a:^  +  2/^  +  ^  =  r^,     ax^  —  by^  —  cz  =  d. 
Eliminating  x  gives 

(a  +  b)y^  +  a^  +  cz  =  ar"^  —  d, 

which  is  the  equation  of  an  ellipse,  the  centre  being  on  the 

axis  of  z  at  a  distance  from  the  origin  equal  to  —  ^.    It  is 

Aa 

the  projection  of  the  curve  on  the  plane  yz. 
Eliminating  y  gives 

{a  +  b)a?  +  b^-cz  =  br'  +  d, 

which  is  also  the  equation  of  an  ellipse  and  is  the  projection 
of  the  curve  on  xz. 

Eliminating  z  gives  an  equation  of  the  form 

Aa^  +  Ba^y^  +  Cy"  +  Da?  +  Ey-  +  F=  0, 

which  is  the  equation  to  the  projection  of  the  curve  on  the 
plane  xy.  This  is  a  curve  of  the  fourth  order,  and,  not  being 
plane,  has  no  special  name. 


192 


CURVED  SURFACES. 


[249. 


24:9.  Intersection  of  a  Plane  and  Cone.  —  Take  a 
riglit  cone  having  a  circular  base  ;  its  equation  will  be, 
(Art.  227), 

'J?  +  'ip  —  ^  tan^v,  (1) 

the  origin  being  at  the  apex.  Transfer  the  origin  to  0,  a 
point  on  the  axis  of  z,  the  distance  ZO  being  c ;  then  will  the 
equation  become 

a;2  +  2/^  =  (2  -  cf  tan^t;.  (2) 

Let  the  secant  plane  embrace  the  axis  of  y,  then  will  it 
be  perpendicular  to  the  plane  xz,  and  BO  will  be  its  trace 

on  that  plane.  Let  the  angle  be- 
tween the  secant  plane  and  the  axis 
of  z  be  u  =  BOZ.  Then  will  the 
equation  of  the  secant  plane  be 
(Art.  203), 

O.y  +  x=  z  tan  v.         (3) 

Eliminating  z  between  equations 
(2)  and  (3),  we  have 

y^  cot^  V  +  (cot-  V  —  cot^  u)a^  +  2cx  cot  u  =  c^,        (4) 

for  the  equation  of  the  projection  on  the  plane  xy  of  the 
curve  of  intersection.  To  find  the  equation  of  the  curve  in 
its  own  plane,  let  P  be  any  point  of  the  curve  whose  coor- 
dinates are 


OD  —  x  =  X  cosec  ti,  DP  =  y'  =  y. 


(5) 


Substitute  the  values  of  x  and  y  from  these  equations  in 
the  preceding  equation,  and  dropping  the  accents  we  have 

y"^  cot^  V  +  sin^  u  (cot^  v  —  cot^  u)oe^  +  2cx  cos  u  —  <y. 

This  is  an  equation  of  the  second  degree  between  two 
variables ;  and,  by  comparing  it  with  the  general  form  given 
in  Article  177,  gives 


250]  OF  INTERSECTIONS.  193 

A  =  sin'^  u  (cot'  V  —  cot"  u);        B  =^  cot^  v ;        H=  0  ; 
G—cGosu;        i^=0;  and  (7=  —  cl 

Hence,  according  to  Article  178,  tlie  curve  of  intersection 
will  be  (since  H  is  zero) 

an  ellipse  if  .  .  .  (cot^  v  —  cot^  u)  is  positive ; 

a  parabola  if    .  .  (cot^  v  —  cot"  u)  =  0  ; 

an  hyperbola  if     (cot^  v  —  cot^  u)  is  negative. 

This  expression  will  be  positive  when  cot  u  <  cot  v,  or 
uy  v;  that  is. 

When  the  secant  plane  makes  a  greater  angle  with  the  axis  of 
the  cone  than  the  elements  do,  the  intersection  is  an  ellipse,  (Art. 
185). 

The  expression  will  be  zero  when  u  =  'V ;  hence, 

When  the  secant  plane  is  parallel  to  an  dement  of  the  cone,  the 
curve  of  intersection  ivill  he  a  parabola. 

The  expression  will  be  negative  when  u  <v;  hence. 

When  the  secant  plane  makes  a  less  angle  with  the  axis  of  tlve 
cone  than  the  elements  do,  the  curve  of  intersection  will  he  an  hy- 
perbola. 

If  the  secant  plane  is  perpendicular  to  the  axis  of  the 
cone,  u  will  be  90°.     The  coefficient  of  x^  is 

sin^  u  cot^  V  —  cos^  w, 

or  for  u  =  90°,  it  becomes 

cot^  Vf 

and  the  equation  of  the  curve  becomes 

y^  +  x'''  =  c^tan^  v, 

which  is  the  equation  of  a  circle. 

250.  To  find  the  intersection  of  a  plane  loith  a  right  cone 
having  an  elliptical  base. 

The  equation  of  the  curve  will  be  of  the  second  degree, 
and  hence  the  curve  will  be  a  conic. 
13 


194  CURVED  SURFACES.  [251-254. 

251.  To  find  the  intersection  of  a  plane  with  a  right  cone 
having  an  hyperbola  for  its  base. 

The  equation  of  tlie  curve  of  intersection  will  be  of  the 
second  degree,  but  if  the  intersection  be  an  ellipse  it  will  not 
be  re-entrant.  ^ 

Similarly,  the  intersection  of  a  plane  with  a  cone  having 
a  parabola  for  a  base  will  be  a  conic. 

252.  To  find  the  intersection  of  a  right  cone  having  an  dlip- 
tical  base  ivith  an  ellipsoid. 

Let  the  equations  be, 

of  the  cone,  ao(?  +by^  =  cz^ ; 

of  the  ellipsoid,         OiO^  +  b^y^  +  Ci^  =  d. 

Eliminating  z  gives 

(ttic  +  aci)x^  +  (bic  +  bci)y^  =  cd, 

which  is  the  equation  of  an  ellipse.  In  a  similar  manner 
find  the  equations  of  the  curve  on  the  other  planes.  It  may 
not  be  a  plane  curve. 

Discussion  of  the  General  Equation  of  the  Second  Degree  having 
Three  Variables. 

253.  A  full  discussion  is  not  here  attempted;  but  some 
of  the  steps  are  indicated  by  which  a  more  complete  discus- 
sion may  be  made.     The  general  equation  is  of  the  form : 

Ax*  +  2nxy+By^  +  2Eyz+Ez*  +  2Lzx  +  2Gx  +  2Fy-h2Dz+C  =  0.       (1) 

This  equation  may  be  divided  through  by  C  (or  any 
other  coefficient)  and  be  equally  general,  after  which  there 
will  be  nine  arbitrary  constants.  This  equation  is  called 
The  Quadric.  Hence,  in  general,  a  quadric  may  be  passed 
through  nine  points  not  in  the  same  plane. 

254.  The  extent  and  character  of  a  surface  may  be  de- 
termined by  intersecting  it  with  planes  and  determining  the 
extent  and  character  of  the  intersections.    Intersecting  a 


255.]  EQUATION  OF  TEE  SECOND  DEGREE.  I95 

Quadric  by  a  plane  will  give  an  equation  of  the  second  de- 
gree; hence  every  plane  section  of  a  Quadric  is  a  Conic. 

If  the  Quadric  be  cut  by  two  planes  parallel  to  xy,  whose 
equations  are  z=  k  and  z  =  k^,  the  curves  of  intersection 
will  be  found  by  eliminating  z  from  equation  (1).  The  two 
resulting  equations  will  contain  the  same  values  of  A,  H, 
and  B ;  hence  parallel  sections  of  a  Quadric  are  similar  Conies. 

255.  Transform  the  coordinates  by  changing  the  direc- 
tion of  the  axes,  the  origin  remaining  the  same.  For  this 
purpose  use  the  equations,  (Art.  219), 

x  =  x  cos  X'  +  y'  cos  X"  +  z  cos  X"\ 
y  =  x'  cos  Y'  +  y'  cos  Y"  +  z  cos  Y'", 
z  =  x  cos  Z'  +  y'  cos  Z"  +  z'  cos  Z'". 

These  values  in  equation  (1)  reduce  it  to  the  form 
A'x^  +2H'xy+B'y^  +  2K'yz  +E  z''  +2L'ex+2Gx  +  2F'p  +  2D'z+l=  0,    (2) 

in  which  the  coefficients  are  functions  of  the  angles.  There 
being  nine  arbitrary  constants,  we  may  make  nine  equations 
of  condition  among  these  coefficients.  Let  the  new  axes  be 
at  right  angles  to  each  other ;  this  gives  the  following  six 
equations,  (Art.  219), 

cos  X'  cosX"  +  cos  Y'  cos  Y"  +  cos  Z'  cos  Z"  =  0, 
cos  X'  cos  X'"  +  cos  Y'  cos  Y'"  +  cos  Z'  cos  Z'"  =  0, 
cos  X"  cosX"  +  cos  y  cos  Y'"  +  cos  Z" cob  Z'"  =  0; 

cos^  X'  +  cos^  Y'  +  cos^  Z'  =1, 
cos^  X"  +  cos^  Y"  +  cos^  Z"  =1, 
cos*  X'"  +  cos^  Y'"  +  cos^  Z'"  =  1. 

We  may  make  three  more  equations  of  condition ;  hence 
we  may  have 

H'  =  0;  K'  =  0',  L'  =  0; 

and  the  equation  reduces,  after  dropping  the  accents,  to  the 
form 

Ax^  +  By^  +  Ez^  +  2Gx  +  2Fy  +  2Dz  +  C  =  0.  (3) 


196  CURVED  SURFACES.  [256,257. 

256.  To  transform  to  paraUd  axes. 

Make         a?  =  a  f  a?',        y  =  b  +  y\        z=  c  +  z\ 

in  equation  (3).  There  being  three  new  arbitrary  constants, 
a,  b,  c,  such  values  may  be  assigned  to  them  as  will  make  the 
coefficients  of  the  first  powers  of  x,  y,  z,  each  equal  to  zero. 
These  values  will  be 

_      G         ;  _       ^  _       D 

which  will  be  real  and  finite  unless  A,  B,  or  E  is  zero ;  that 
is,  they  will  be  real  when  the  equation  contains  the  second 
powers  of  all  three  variables.  When  such  is  the  case  the 
equation  becomes 

Ax'^  +  By'^+Ez'  +  (7=0.  (4) 

In  this  equation,  ii  —  x  be  substituted  for  x,  —  y  for  y, 
and  —z  for  z,  neither  the  form  nor  value  of  the  equation 
will  be  changed ;  hence,  every  line  drawn  through  the  origin 
and  terminated  by  the  surface  is  bisected  at  the  origin. 
The  origin,  therefore,  is  at  the  centre  of  the  surface.  Such 
quadrics  are  called  centkal  quadrics. 

257.  The  Central  Quadric. — When  the  equation  of  the 
central  quadric  is  reduced  to  the  form 

Ax"  +  By''  +  Ez'+G^O;  (5) 

the  axes  are  called  Principal  Axes,  and  the  sections  made  by 
the  coordinate  planes,  Principal  Plines.  It  has  been  shown 
that  this  transformation  is  always  possible  for  central  quad- 
rics ;  hence,  conversely.  Every  central  quadric  has  at  least  one 
set  of  three  conjugate  planes  and  three  diameters  which  are  mutu- 
ally perpendicular. 

If  A,  B,  and  E  are  positive,  and  C  negative,  we  have 

Ax^  +  Bf  +  Ez'=C; 

which  is  the  general  equation  of  the  ellipsoid^  (Art.  233). 


258.]  EQUATION  OF  THE  SECOND  DEGREE.  I97 

Jl  A=  B,  it  is  the  equation  of  tlie  ellipsoid  of  revolution. 

If  A—  B  =  E,  it  is  the  equation  of  tlie  sphere. 

If  (7—0,  then  for  a  real  locus  x  =  0,  y  =  Q,  z  =  0,  which 
are  the  equations  of  a  point.  If  x,  y,  z,  are  not  zero,  it  is 
the  equation  to  an  imaginary  ellipsoid. 

If  C  is  positive,  the  surface  is  imaginary. 

Again,  if  two  of  the  coefficients  A,  B,  E,  are  positive 
and  one  negative,  let  A  and  B  be  positive  and  E  and  C  neg- 
ative ;  then  we  have 

Ax"  +By^  -Ez"^  C; 

which  is  the  equation  of  an  elliptical  hyperholoid,  (Art.  240). 

If  A  =  B,  it  is  the  equation  of  an  hyperholoid  of  revolution 
of  one  nappe. 

If  ^  =  ^  =  ^,  it  is  the  equation  of  the  equilateral  hyper- 
holoid of  revolution  of  one  nappe. 

If  (7=0,  it  is  the  equation  of  the  surface  of  a  right  cone, 
(Art.  227). 

If  C  be  positive,  the  equation  becomes 

-Ax'-By''  +  Ez'=C, 

which  is  the  equation  of  the  hyperholoid  of  tivo  nappes.  If  A, 
=  B,  it  is  an  hyperholoid  of  revolution,  and  if  A  =  B  =  E, 
it  is  an  equilateral  hyperholoid  of  revolution  of  two  nappes. 

258.  Suppose  that  one  of  the  coefficients  A,  B,  E,  as  A, 
is  zero  in  equation  (3).  Again  transform  the  origin,  and  de- 
termine the  values  of  the  new  constants  by  making  the  co- 
efficients of  z  and  y,  and  the  absolute  term,  separately  equal 
to  zero,  and  we  will  find 

which  is  the  equation  of  a  paraboloid.     It  is  a  non-central 

QUADRIC. 

If  B  and  E  are  positive  and  G  negative,  it  will  be  an  ellip- 
tic paraboloid,  (Art.  235).  li  B  =  E  and  G  negative,  it  is  a 
paraboloid  of  revolution,  (Art.  234). 

If  B  is  negative  and  E  positive,  or  the  reverse,  it  is  an 
hyperbolic  parahdoid,  (Art.  236). 


198  CURVED  SURFACES.  [259-261. 

259.  The  preceding  transformation  is  impossible  if  all 
the  terms  containing  x  are  zero  ; — that  is,  if  ^  =  0,  and  O 
=  0.     These  reduce  equation  (3)  to 

B^f  +  E^  +  2Fy  +  2I)z+  C  =  0, 

which  is  the  equation  of  a  cylindrical  surface  whose  axis  is 
parallel  to  the  axis  of  x  and  whose  base  is  a  circle,  ellipse, 
or  hyperbola,  depending  upon  the  relative  values  and  signs 
of  B  and  K 

260.  If  ^  =  0  and  B  ^0,  equation  (3)  becomes 

Ezl  +  2Gx  +  2Fy  +  2Dz  +  C  =  0. 

Intersecting  this  surface  by  planes  parallel  to  xy,  the  equa- 
tion of  one  of  which  will  he  z  =]c,  we  have 

2Gx-\-2Fy  =  K{BSij),  ^ 

which  is  the  equation  of  a  straight  line.  Hence  the  ele- 
ments are  parallel  straight  lines,  parallel  to  the  plane  xy. 
Making  x  =  0,  we  have 

Ez^-{-2Fy  +  2Dz+C--:=0, 

which  is  a  parabola,  and  is  the  intersection  of  the  plane  zy 
with  the  surface.     Similarly,  making  y  =  0,  we  have 

Ez''+2Gx  +  2Dz+C  =  0, 

which  is  also  the  equation  of  a  parabola,  and  is  the  intersec- 
tion of  the  plane  zx  with  the  surface.  The  surface,  there- 
fore, is  a  cylinder  having  a  parabolic  base. 


Of  Tangent  Planes. 

261.    To  find  the  equation  of  a  plane  tangent  to  a  centred 
quadric. 

The  general  equation  of  the  surface  is 

Aa?+By'  +  Ez'-{-C=0.  (1) 


361.]  OF  TANGENT  PLANES.  199 

Let  the  point  of  tangency  be  (x,  y ',  z'),  and  we  have  the 
equation  of  condition 

Ax'  -  +  %'"-  +  j^fe'  -  +  0  =  0,  (2) 

which  subtracted  from  the  preceding  equation  gives 

A{,i?  -  X-)  +  B{y-  -  y'  -)  f  E{z'  -  z"")  ^  0.         (3) 

The  equation  of  a  plane  passing  through  the  point  of 
tangency  is 

a{x  —  x)  +  h{y  —  y')  +  c{z~  z)  =  0,  (4) 

in  which  such  values  must  be  substituted  for  a,  h,  c,  as  will 
make  the  plane  tangent  to  the  surface. 

If  two  lines  be  passed  through  the  point  tangent  to  the 
surface,  the  plane  of  these  lines  will  be  the  tangent  plane 
required.  Let  a  plane  be  passed  through  the  tangent  point 
parallel  to  a;2 ;  it  will  cut  a  section  from  the  quadric  and  a 
secant  to  the  section  from  the  cutting  plane.  The  equation 
of  the  plane  will  be 

which  substituted  in  equations  (3)  and  (4)  will  give 

A  {x  +  x')  {x  -  x')  +  E {z  +  z')  (z  -  z')  =  0, 

and  a{x  —  x')  +  c{z  —  z')  =  0. 

The  value  of  (x  —  x)  from  the  last  equation  substituted  in 
the  preceding  gives 

A  X  -\-  x' 

E  z+  z' 

Let  the  secant  turn  about  {x,  z)  until  the  point  {x, z)  co- 
incides with  {x'yz),  then  will  the  secant  become  a  tangent, 
and  we  have 

Ax' 


200  CURVED  aUBFACEa.  [262,263. 

Similarly,  we  may  find 

and  these  values  substituted  in  equation  (4),  and  the  result 
reduced  by  means  of  equation  (2),  give 

Axx'  +Byy'  +  Ezz'  +  (7=0, 

for  the  required  equation. 

262.  To  find  the  equation  of  a  plane  tangent  to  a  non-central 
quadric. 

The  equation  of  the  surface  is 

By-  ^-Ez'  +  '^Gx^O. 
Proceeding  as  before,  we  find 

Byy'  +  Ezz'+  G{x+  x')  =  0, 
for  the  required  equation. 

Of  Normals  to  Quadrics. 

263.  A  normal  is  a  line  perpendictdar  to  a  tangent  plane  at 
the  point  of  contact.  Hence,  the  equations  of  the  normal  for 
central  quadric  are,  (Art.  214), 

,     Ax'  ,      By',       ,. 

«^-^  =  ^^i^-^)>      y-y  =  jf-(^-0» 

and  for  paraboloids,  are 

[For  a  farther  discussion  of  Quadrics,  see  Salmon's  Geometry  of  Three 
Dimemions.'^ 


263.]  NORMALS  TO  QUADBICS.  201 


EXAMPLES. 

1.  Determine  the  class  of  surfaces  to  wliich  the  following  quadrics 
belong : 

7^■-  +  6?/-  +  52-  —  Ayz  —  Axy  =  6, 
?>x-  —  4^-  +  ~z'-  —  xy  +  "Zyz  —  ^xz—  12, 
^x^  -  2^2  _  32  +  4^;  =  5, 
2x^  +  %tf  +  4^2  =  0.  * 

2.  Tangent  planes  at  the  extremities  of  any  diameter  are  parallel. 

3.  In  any  central  quadric  the  sum  of  the  squares  of  three  conjugate 
diameters  is  constant. 

4.  The  locus  of  the  intersection  of  three  planes  tangent  to  an  ellip- 
soid which  are  mutually  perpendicular,  is  the  sphere 

2;8    +    2/2    +  S2  =  a2    +  &*  +  C^. 


CHAPTEE  Yin. 

LOCI     OP     HIGHER     ORDERS. 

264.  Definitions. — Higher  plane  curves  include  all  loci 
whose  equations  cannot  be  reduced  to  the  first  or  second 
degree.  In  earlier  works  upon  this  subject  they  were 
divided  into  the  two  classes,  Algebraic  and  Transcendental; 
which  classification  would  be  proper  if  bilinear  coordinates 
only  were  used ;  but  in  Modern  Geometry  there  are  numer- 
ous systems  of  coordinates,  and  the  same  curve  may  be  ex- 
pressed algebraically  in  one  system,  or  transcendentaUy  in 
another.  Thus,  for  example,  the  circle  is  expressed  alge- 
braically by  the  equation 

and  transcendentaUy  by  the  equation 

sin  £  =  a  cos  [V—  1  (log  r  +  &)], 

in  which  r  denotes  the  radius  vector  of  the  curve,  f  the  angle 
between  r  and  the  tangent  at  the  point,  a  and  h  constants. 
(See  Math.  3Ionthhj,  1858,  pp.  11  and  58.) 

When  curves  are  classed  as  transcendental,  it  is  implied 
that  their  equations  involve  trigonometrical,  circular,  loga- 
rithmic, or  exponential  quantities.  Algebraic  curves  are 
such  as  may  be  expressed  by  algebraic  quantities.  They  are 
classed  according  to  the  degree  of  the  equation;  thus,  an 
equation  of  the  second  degree  represents  a  curve  of  the  sec- 
ond order ;  of  the  third  degree,  the  third  order ;  and  of  the 
nih.  degree,  the  nih  order. 

202 


265,  266.] 


OF  SPIBALS. 


203 


The  number  of  higher  plane  curves  is  unlimited.  It  is 
known  that  there  are  at  least  eighty  species  of  lines  of  the 
third  order,  and  more  than  5000  sjoecies  of  the  fourth  order. 
Only  a  few  of  the  higher  curves,  and  those  most  noted,  will 
here  be  considered. 

Of  Spirals. 

265.  A  Spiral  is  a  curve  which  may  be  generated  by  a 
point  moving  uniformly  around  a  fixed  j)oint  and  whose  dis- 
tance from  the  fixed  point  varies  according  to  an  assigned 
law. 

The  fixed  point  0  is  called  the  ^ 

pole  of  the  spiral.  y-'\  /^\ 

A  spire,  or  whorl,  is  a  portion 
of  the  spiral  generated  during 
one  revolution  of  the  generating 
point.  Thus  L3INE  is  one-half 
of  a  spire. 

The  measuring  circle  is  one  de- 
scribed with  a  radius  unity,  hav- 
ing the  pole  as  the  centre.    Thus, 
if  Oi?  be  a  unit  radius,  then  will  the  circle  BCD,  etc.,  be  the 
measuring  circle. 

The  radius  vector  is  the  distance  from  the  pole  to  any 
point  of  the  curve,  as  OL,  OM,  etc. 

266.  The  Spiral  of  Archimedes  is  a  curve  which  maybe 
generated  by  a  point  moving 
uniformly   along   the   radius 
vector  while  that  radius  has 
an  uniform  rotary  motion. 

To  construct  it,  divide 
the  measuring  circle  BGD^ 
etc.,  into  equal  parts,  BC  = 
CD  —  DE,  etc.,  and  draw 
radial  lines  OB,  00,  etc.,  and 
let  OX  be  the  polar  axis  (or 
initial  line).  Assume  OJ  (or  find  it  from  given  conditions), 
and  make  OK^  WJ,  OL  =  30/,  etc.,  then  will  J,  K,  L,  etc., 
be  points  in  the  spiral. 


Fui.  175. 


Fm.  176. 


204 


HIGHER  LOCI. 


[2G7 


To  find  the  equation  to  the  locus,  let  6  be  the  variable 
angle,  measured  by  the  arc  BCD,  etc.,  r  the  radius  vector, 
a  the  ratio  of  the  radius  vector  to  the  variable  angle  ;  then, 
according  to  the  definition,  we  have 

r  =  ad,  ^  (1) 

which  is  the  required  equation. 

If  the  radius  vector  OB  at  the  end  of  the  first  spire  be 
taken  as  unity,  we  have 


1  —  a.27r 
and  equation  (1)  becomes 


a  = 


27r' 


r  = 


27r' 


(2) 


If  0  =  0,  r  =  0,  hence  the  curve  passes  through  the  pole.  If  0  be  neg- 
ative, r  will  be  negative,  and  the  locus  will  be  the  same  as  for  0  positive. 
r  increases  directly  with  0  from  zero  to  infinity. 

For  one  whorl  r^  =  1, 

for  two  whorls  r^  =  2  ; 

.-.  rg  — r,  =  1  =ri  ; 

and  generally,  the  radial  distance  between  any  two  consecutive  whorls  is 
constant,  and  equals  the  radius  at  the  end  of  the  first  whorl. 

Example. — A  string  is  wound  spirally  around  a  cone,  extending  from 
the  apex  to  the  base,  dividing  the  slant  height  into  n  equal  parts,  the  radius 
of  the  base  being  i2,  and  the  slant  height  I.  The  cone  is  placed  on  a  plane 
and  rolled  in  such  a  way  as  to  unwind  the  string,  the  string  remaining  on 
the  plane  as  it  is  unwound  ;  required  the  equation  of  the  curve  of  the  string. 

267.  The  Reciprocal  or  Hyperbolic  Spiral  is  a  curve 

in  which  the  radius  vector  va- 
ries inversely  as  the  measur- 
n-- — ::^<C  /i.  ■    ing  arc. 

Its  equation  is 


a 

T  =  — . 

0 


(1) 


of 
have 


If   the    radius  at  the  end 
one   whorl    is    unity,     we 


268.] 


OF  SPIRALS. 


205 


l=^;.'.a=27t. 


and  the  equation  becomes 


r  — 


27r 
d  ' 


(2) 


If  Q  =r  0,  r  =  00.  As  0  increases,  r  diminishes,  and  wlien  0  =  27r,  r  =  1, 
so  that  the  radius  vector  passes  from  1  to  oo  in  one  whorl.  If  r  =  0,  0  =  oo, 
hence  there  are  an  infinite  number  of  whorls  between  the  pole  and  the  measur- 
ing circle. 

The  two  preceding  spirals  are  special  cases  of  the  curve 


r  =  a'j^. 


in  which  n  may  be  +1  or  —1. 


268.  The  Logarithmic  or  Equiangular  Spiral  is  de- 
fined by  the  equation 

ad  =  log  r,  or  r  —  e"^ ; 


from  -which  it  follows  that  the  D^.- -^--C 

logarithm  of  the  radius  vector  is 
proportional  to  the  measuring 
arc. 

If  r=  l=OB,e=0;  hence  if  r 
=  1  be  initial,  the  measuring  arc 
will  begin  at  B  on  the  polar  axis. 

If  r  =  0,  0  =  —  oc;  hence,  be- 
tween the  pole  and  r  =  1,  there 
will  be  an  infinite  number  of  whorls. 

The  curve  may  also  cross  the  polar  axes  an  infinite  number 
of  times  for  values  of  r  greater  than  unity ;  for  r  will  be 
real  for  values  of  6  =  27r,  Att,  Qtt,  etc. 

To  construct  the  curve  make  the  measuring  arcs  BC  = 
CD  =  DE,  etc.,  and  lay  off  OL,  031,  ON,  etc.,  in  geometrical 


Fia.  178. 


progression;  that  is, 


OM     ON 


,  etc. 


OL  ~  OM 
This  curve  is  called  equiangular  because  the  tangent  at 


206  HIQHEB  LOCI.  [3G9. 

any  point  makes  a  constant  angle  witli  the  radius  vector  at 
that  point.* 

269.  The  Involute  of  a  Circle  is  the  locus  of  any 
7^     point  of  a  line  as  it  rolls  upon  a  circle. 

Thus,  let  p  be  any  point  of  ^the  line  Ap,  then  if 
Ap  be  rolled  upon  the  curve,  F  will  describe 
the  arc  pB,  and  if  it  continues  to  roll  in  that 
direction,  it  will  describe  a  second  branch  of 
the  curve.  The  SbVcpB  is  the  involute  of  the 
circle  whose  centre  is  0  and  radius  OA,  The 
same  curve  may  be  described  by  the  end  of 
a  string  as  it  is  unwound  from  the  circle.  In  a  similar  man- 
ner the  involute  of  any  other  curve  may  be  described. 

To  find  an  equation  to  the  involute  of  the  circle,  let  r  — 
OA,  p  =  Op  =^  the  radius  vector,  Ap  =  arc  AB  =  rB  ;  then 
since  Ap  is  tangent  to  the  circle,  the  right  triangle  pAO 
gives 

p^  =  r-+  {rdf  r=  r\l  +  ^), 

which  is  the  required  equation  in  one  form ;  but  the  system 
to  which  it  is  referred  is  peculiar.  To  find  the  equation 
referred  to  rectangular  coordinates,  take  the  origin  at  the 
centre  0,  the  axis  of  x  passing  through  the  point  B  and 
positive  from  0  towards  B,  and  6  positive  from  B.  Then 
we  may  find  from  the  figure  that 

X  =r  cos  6  +  rd  sin  6, 
y  =  r  sin  0  —  rd  cos  6 ; 

*  [Differentiating  the  equation  to  the  curve,  gives 
— -=  ae^9  —ar; 

da 

. '.  —37;  =  a=  a  constant. 
rdB 

But  the  first  member  is  the  tangent  of  the  angle  between  the  normal  and 
the  radius  vector ;  hence  t?ie  curve  cuts  the  radius  vector  at  a  constant  angle. 

Example. —  An  equiangular  spiral  whose  equation  is  aO  =  logr,  rolls  on 
a  straight  line  ;  required  the  locus  of  the  pole. 

Ans.  A  right  line  cutting  the  given  line  at  an  angle  whose  tangent  w  a.] 


270,271] 


OF  SPIRALS. 


2C7 


and  if  0  could  be  eliminated  from  these  three  equations  we 
should  find  the  rectangular  equation  to  the  locus ;  but  if 
eliminated  the  result  would  contain  circular  functions,  and, 
therefore,  the  equation  would  be  transcendental. 

{Rem. — The  involute  is  often  used  in  the  construction  of  teeth  in  gearing. 
The  length  of  the  involute  of  the  circle  is   given  by  the  expression  Bp  = 

^  rO^ ;  and  the  area  ABp  —  -  r^O^.) 
«  6 

270.  The  lituus  is  a  curve  defined  by  the  equation 

/3-(3  =  a-. 
If  p  =  0,  0  =  00  ; 
if  6  =  0,  p  =  00. 
The  equation  may  be  written 


a 


Fig.  180. 


hence  there  are  two  equal  values  of  p  corresponding  to  each  value  of  0,  one 
of  which  is  positive  and  the  other  negative. 
If  /3  =  1  when  0  —  2n,  then 


1  = 


V2,Tt 


=  V27C. 


and  the  equation  becomes 

p2G  =  Stt. 

It  is  found  by  higher  analysis  that  the  initial  line  OX  is  an  asymptote 
to  the  curve. 

271.  Parabolic  Spiral. — If  the  axis  of  a  parabola  whose  equation  is 
y^  =  lipx,  be  wrapped  around  the 
circumference  of  a  circle,  and  the 
corresponding  ordinates  of  the  pa- 
rabola be  laid  off  on  the  radius  pro-  /CB 
longed,  the  locus  will  be  a  parabo- 
lic spiral.  Let  p  =  CB,  r  =  CA 
=the  radius  of  the  circle,  then  will 
the  ordinate  y  be 

y  =  p  —  r, 


iai  the  corresponding  abscissa, 

and  these  in  the  equation  of  the  parabola  give 
(/>-r)«=2rpe. 


Pie.  181. 


208  HIGHER  LOCI.  [272,  273. 

which  is  the  required  equation.     Let  p  ~  nr,  then  the  equation  becomes 
(71  -  1)2  r  =  2p^. 

T 

If  7i  =r  0,  p  =  0,  and  G  =  -g— ,  which  being  real  shows  that  the  locus 

passes  through  the  centre  of  the  circle.     As  n  increases  Gi  decreases  until 

re  =  1,  for  which  value  0  =  0,  and  p  =  r.     This  is  the  initial  point  of  the 

curve,  from  which  point  G  increases  as  p  increases.     If  n  (.or  r)  be  negative, 

2r 
6  increases,  and  when  p  —  —  r  we  have  n=  —  \,  and  G  =  -.  ,  at  which  point 

the  locus  again  crosses  the  circumference,  and  G  \vill  increase  as  r  increases 
negatively. 


form 


Trigonometrical  Curves. 

272.  Trigonometrical  curves  are  such  as  involve  a  trigo- 
nometrical expres- 
sion in  the  equations 
of  the  curve. 

273.  The  Sinu- 
soid is  a  curve  whose 
equation  is    of    the 


a  sm 


&' 


(1) 


in  which  a  and  h  are  constants.     If  a  and  h  are  each  unity, 
the  equation  becomes 


y  =  sm  X, 


(2) 


which  is  called  the  equation  of  sines,  and  the  correspond- 
ing curve  is  called  the  curve  of  sines. 
To  construct  the  sinusoid,  let 

a;  =  ^6;r  =  Jl,  then  y  =  asin^V  ^J 

x  =  r^h7r  =  A%  y=asm^n:; 

x=r^b7T=:A3,  y  =  asmj\7r; 

etc  etc 


274-378] 


LOGAIilTHMIC  CURVES. 


209 


Erecting  ordinates  at  the  points  1,  2,  3,  etc.,  equal  to  the 
corresponding  values  of  ij,  gives  points  in  the  curve. 

274.  Remark.— The  sinusoid  is  used  in  Physics  to  express  certain  laws 
of  motion.  It  expresses  the  law  of  movement  of  the  vibration  of  perfectly 
elastic  solids  ;  of  the  vibratory  movement  of  a  particle  acted  upon  by  a  force 
which  varies  directly  as  the  distance  from  the  origin  ;  approximately,  the 
vibratory  movement  of  a  pendulum  ;  and  exactly  the  law  of  vibration  of 
the  so-called  mathematical  pendulum. 

275.  The  curve  of  Tangents  is  expressed  by  the  equation 

y  =  tan  x. 

If  a;  =  ^  TT,  3^  =  oo, 

x=0,  Tf,  2Tt,  etc.,  y  =  0. 


The  curve  begins  at  the  origin  A.  The  or- 
dinate DC,  whose  abscissa  is  {n,  is  an  asymptote 
to  the  curve.  EF  is  another  asymptote,  etc. 
The  curve  has  an  infinite  number  of  branches. 


Fig.  183. 


276.  The  curve  of  Secants  is  expressed  by  the  equation 


y  =  sec  X. 
It  x  =  0,y  =  \  =A\. 

If  X  =  -\it,  y  =  (x>,  and  DC  is  an  asymptote. 
Similarly,  EE,  at  a  distance  ^tc  from  the  origin  is  an 
asymptote.  If  x  =  tt,  then  y  =  —  1  ;  if  x  =  2n, 
then  y  =  1  =  Bl'.  This  curve  also  has  an  infinite 
number  of  branches. 

277.  In  a  similar  manner  curves  may  be  found 
for  each  of  the  remaining  five  trigonometrical  func- 
tions.    Circular  functions  give  corresponding  curves. 


J 

jl 

C 

F 

\ 

A 

A 

1 

B 

D 

/ 

\ 

E 

Fig.  184. 


Logarithmic  Curves. 

278.  A  liOgarithmic  Curve  is  one  in  which  the  abscissa 
is  the  logarithm  of  the  ordinate,  or  the  ordinate  is  the  loga- 
rithm of  the  abscissa.     Its  equation  is 


X  =  log  y. 

If  a  be  the  base  of  the  system  of  logarithms,  its  equa- 
14 


210 


HIGHER  LOCL  [279.  280. 

E       tion  may  be  written 

y  =«''• 

lfa;=0  wefindy=l=^5; 

x=l—Alf        "      "  y—a^=lb; 
x=2=A2,  "      "   y=a^=2c; 

etc.  etc. 

a;=  — 1  =— IJ.,  "      "  y=a-^  =z—ld; 
X—  —  00,  "      "  y—0. 


Fig.  185. 


The  ordinate  at  the  origin  will  always  be  unity  of  the 
scale  on  which  the  locus  is  constructed,  but  the  other  ordi- 
nates  will  depend  upon  the  value  of  the  base  of  the  system. 
The  curve  extends  indefinitely  in  both  directions  from  the 
origin,  and  the  axis  of  x  is  an  asymptote  to  the  curve  on  the 
negative  side  of  the  origin. 

If  the  base  is  unity,  the  locus  will  be  parallel  to  the  axis 
of  X,  and  the  ordinates  cannot  express  a  series  of  numbers; 
hence  the  base  of  a  system  of  logarithms  cannot  be  unity. 


Of  Parabolas. 

279.  All  curves  expressed  by  the  equation 

y  =  mx'^ 

are  called  parabolas.  The  values  of  n  and  m  may  be  fractional 
or  entire,  and  positive  or  negative ;  but  we  shall  here  con- 
sider m  as  positive. 

If  ic  =  0,  2/  =  0,  hence  all  these  curves  pass  through  the 
origin. 

280.  The  Common  Parabola. — Making  «  =  ^,  we  have 

y^  =  rri^x^ 

which  is  the  equation  of  the  common  parabola,  m'  being  the 
parameter,  (Art.  86). 


281-284.  J 


OF  TROCHOIDS. 


211 


281.  The  Cubical  Parabola.— Makins 


y  —  mx^, 


•which  is  called  the  equation  to  the  cn- 
bical  parabola.  It  extends  indefinitely 
to  the  right  of  the  origin  and  above 
the  axis  of  a;  and  to  the  left  of  the 
origin  and  below  the  axis  of  x  ;  and  is 
convex  to  that  axis. 


— X 


li  n  =  ^  we  have 


yS     _     ^3^^ 


which  is  the  equation  of  a  cubical  parabola  convex  to  the 
axis  oiy. 

282.  The  Semi-Cubical  Parabola. — Making  /i=t,  we 
have 

y  =  mx-^,  /° 

which  is  called  the  equation  to  the  semi- 
cubical  parabola.  It  extends  indefinitely 
to  the  right  of  the  origin  and  above  and 
below  the  axis  of  x.  It  is  convex  to  that 
axis,  and  symmetrical  in  reference  to  it. 
If  «  =  I,  we  have 


Fis.  187. 


y^  =  m-^x, 


which  is  a  semi-cubical  parabola,  convex  to  the  axis  of  y. 


283.  The  Biquadratic  Parabola  is  given  by  the  equa- 


tion 


y  =  mx*,  or  y*  —  m*x, 


and  is  deduced  from  the  general  equation,  (Art.  279),  by 
making  w  =  4,  or  n  =  I.  These  curves,  in  their  general 
shape,  resemble  the  common  parabola,  the  former  being  con- 
cave to  the  axis  of  y,  and  the  latter  concave  to  the  axis  of  ./*. 

Of  Trochoids. 

284.  A  Trochoid  is  the  locus  of  a  point  in  a  circle  roll- 
ing upon  a  line.     The  generating  point  may  be  within  the 


212 


HIGHER  LOCI. 


[285. 


circle,  on  its  circumference,  or  entirely  without ;  and  tlie  di- 
rectrix may  be  a  right  line  or  a  curve. 

285.  Tlie  Cycloid  is  a  trochoid  in  which  the  path  is  de- 
scribed by  a  point  in  the 
circumference  of  a  circle  roll- 
ing on  a  straight  line.  Thus, 
if  the  circle  CP  rolls  on  the 
straight  line  AX,  the  point 
P  will  describe  the  arc  of  a 
cycloid  APBX. 

To  find  its  equation,  take  the  origin  at  A,  C  the  cen- 
tre of  the  circle,  r  =  CE,  P  any  point  in  the  curve,  and  we 
have 

x  =  AD,    y  =  PD  =  LE. 


r' 


But  AE  =  arc  PE  —  r  vers ' 

Also    AD=AE-  DE,  and  DE=^PL=  ^ HL .  LE ; 

.'.  x=r  vers" ^  ^  —  DE , 
r 

.1  y  ' 


=  r  vers     ^  -  ^2ry  -  f 


(1) 


which  is  the  required  equation. 

If  y  —0,  X  =  0,  ±  27r,  ±47r,  etc.;  hence  the  curve  has  an 
infinite  number  of  branches  above  the  axis  of  x. 

li  y  ^  2r,  x  =  7rr  =  AF=  lAX. 

If  the  origin  be  at  B,  BM  =  ?/,  and  3IP  =  x,  the  equation 
becomes,  by  changing  x  to  nr  —  x,  and  y  io2r  —  y,  (Art.  49), 

y 


X 


rvers"  -  +  \/2ry—y\ 


(2) 


Draw  a  radius  PC,  and  let  PCE=  <p,  =  the  angle  de- 
scribed by  the  radius  of  the  circle,  while  the  point  P  de-- 
scribes  the  arc  AP,  then,  the  origin  being  at  A,  we  find 

AE  =  arc  PE  —  rep  \ 

x  =  r{rp—  sin  fp)     > ,  (3) 

y  =  r(l—  cos  qj)     J 


286-288.] 


OF  TROCHOIDS. 


213 


which  are  equations  of  the  curve  in  terms  of  <?>,  and  are  often 
convenient  in  solving  problems. 

Kemakks. — The  curve  of  quickest  descent  of  a  body  from  one  point  to 
another  down  a  smooth  surface,  is  a  cycloid.  The  involute  of  a  cycloid  is  an 
equal  cycloid  in  another  position.  (This  is  proved  by  higher  analysis.)  The 
cycloidal  pendulum,  in  which  the  pendulum  describes  the  arc  of  a  cycloid, 
is  of  historical  interest,  but  is  not  considered  of  much  practical  value. 

286.  The  Prolate  Cycloid  is  the  path  described  by  a  point  within  the 
circle  rolling  on  a  straight 
line. 

Let  P  be  the  generating 
point  at  a  distance  from  the 
centre  =  PC  =  b,  0  the  ori- 
gin, 

r  =  CR,<p  =  PCB,  X  ■=  OD, 

then  x  —  rq)  —  b  sin  q), 

y  —  r  —  h  cos  q),  (4) 

are  the  equations  of  the  locus,  in  which  h  <r. 

287.  The  Curtate  Cycloid  is  the  path  described 
by  a  point  without  the  rolling  circle.  The  equa- 
tions are  the  same  as  in  the  preceding  Article, 
excepting  that  we  must  make  b>  r.  If  6  =:  r,  we 
have  the  cycloid.  Fig.  190, 

288.  The  Hypercycloid  (or  Epicycloid)  is  the  path  traced  by  a  point 
in  the  circumference  of  the  generating  circle  as  it  rolls  on  the  convex  side 
of  another  circle. 

Let  B  be  the  radius  of  the  direc- 
trix ;  r  the  radius  of  the  generatrix, 
6  =  CO  A,  (p  =  BCP,  P  the  point  in 
the  curve  AP. 

Then  arc  BA  =  arc  BP,  or 

BB  =rg3;  .-.  <p=  —0. 

The  inclination  of  CP  to  the  axis 

of    X  is  0  +  (p,  or    substituting  the 

r  +  B 


value  of  <p,  it  becomes 


The 


Fio.  191. 


projection  of  CO  on  the  axis  of  x  will  be  (B  +  r)  cos  0 ;  and  of  CP,  r  cos 


rJrB 


214 


HIGHER  LOCI. 


[289.  290. 


hence  the  abscissa  of  P  will  be 


R 


a;  =  (5  -H  r)  cos  9  —  r  cos  • 

i2+  r 
and  similarly,      y  =  (i2  +  ?•)  sin  0  —  r  sin  — - — 

which  are  the  equations  of  the  curve  in  terms  of  Jhe  angular  movement  of 
the  line  of  centres.  If  R  and  r  are  incommensurable,  the  curve  will  have 
an  infinite  number  of  branches,  but  if  they  are  commensurable,  the  curve 
will  repeat  itself.  When  they  are  commensurable  6  may  be  eliminated,  and 
a  single  algebraic  equation  found  for  the  locus. 

259.  The  Hypertrochoid  is  the  path  traced  by  any  point  on  the  radius 
of  the  generating  circle  as  it  rolls  on  the  convex  arc  of  a  fixed  circle. 


Fig.  192. 

Let  6  be  the  distance  of  the  generating  point  from  the  centre  of  the  gen- 
erating curve,  and  the  other  notation  as  in  the  preceding  Article;  then  we  find 

R 

x  —  (R  +  r)  cosQ  —  &  cos  — 

y  ■={R  +  r)  sin  9  —  6  sin 
for  the  equations  of  the  curve. 


R-^r 


(6) 


Pia.  193. 


290.  The  Hypotrochoid  is  the  path 
traced  by  any  •point  in  the  radius  of  the 
generating  circle  rolling  on  the  concave 
arc  of  another  curve. 

Let  R  =  OA  =  the  radius  of  the  gen- 
erating circle,  r  =  CB  —  the  radius  of 
the  moving  circle,  h  —  CP  =  the  dis- 
tance of  the  generating  point  P  from 
the  centre  C  (P  being  on  the  dotted  line), 
9  =  BOA,  (p  =  BCP  --  the  angle  through 
which  the  generating  circle  has  revolved 
from  the   initial  point ;    then  will  <p  = 


291,  292.] 


OF  TROCHOIDS. 


215 


e  +  supplement  of  the  inclination  of  CP  to  the  axis  of  x.    Also 

R 

arc BA  =  arc BP ;  .-.  q>  =  —  6  ; 


inclination  of  CP  =  q)  —  li  =z 


R 


and  the  equations  to  the  curve  become 

X  —  {R  —  r)  cos  0^5  cos 


R 


R-  r 
y  =  {R  —  r)  sin  0  —  b  sin  — - —  0 


(7) 


291.  The  H3rpocycloid  is  the  path  traced  by  a  point  on  the  circumfer- 
ence of  the  generating  circle  rolling  on  the  concave  arc  of  the  fixed  circle. 

Hence  the  equations  of  the  curve  are  formed  by  making  6  =  r  in  equa- 
tions (7).     They  are 

x  =  {R  —  r)  cos  0  +  r  cos 9  , 


R-r 
g  =  {R  —  r)  sin^  —  r  sin 0 . 


(8) 


The  curve  is  represented  by  the  full  line  passing  through  A. 

292.  Four  Cusped  Hypocycloid. — Let  r  =  \R,  then  will  the  curve 
consist  of  four  branches,  and  form  four  cusps. 
Making  r  =  \R  in  equations  (8),  we  have 

x  =  3r  cos  S  +  r  cos  30, 

y  =  3r  sin  G  -  r  sin  35.  (9) 

From  Trigonometry  we  have 

cos3Q  =4cos*e  -3 cos 9, 

8in39  =  3sinO  -  4sin3  9, 

which  substituted  give 

X  =  Rcos^  0, 

y  —  R  sin'  9  ; 

.-.  a;^  =  B^  cos-  0  ;        y^  =  R^  sin*  9. 

Adding,  observing  that  sin-  9  +  cos*  9  =  1,  wo  have 

which  is  the  equation  to  the  curve. 


Fig.  191. 


(10) 


Remarks. — Hyper  is  from  the  Greek  and  means  over  or  above ;  and 
Hypo,  under.  The  hypercycloid  and  hypocycloid  are  often  used  in  the  theo- 
retical construction  of  the  teeth  of  gear  wheels.     In  the  hypercycloid,  if  the 


216  HIGHER  LOCI.  [393-295. 

radius  of  the  fixed  circle  be  infinite,  the  curve  becomes  a  cycloid.  If  the 
radius  of  the  generating  circle  be  infinite,  the  hypercycloid  becomes  an  in- 
volute. If  the  diameter  of  the  generating  circle  equals  the  radius  of  the 
fixed  circle,  the  hypocycloid  becomes  a  straight  line  and  will  be  the  diame- 
ter of  the  fixed  circle,  and  any  trochoid  will  be  an  ellipse.  If  the  radius  of 
the  generating  circle  equals  the  diameter  of  the  fixed  circle,  and  the  gener- 
ating circle  be  conceived  to  roll  within  the  fixed  one,  the  centre  of  the  gen- 
erating circle  will  describe  the  circumference  of  the  fixed  one,  and  any 
point  on  the  radius  will  describe  a  curve  called  the  Lima<;on  (see  next  Aiticle>, 
and  any  point  on  the  circumference  a  Cardioid.  If  the  diameter  of  the 
generating  circle  equals  that  of  the  fixed  one,  and  rolls  outside  the  fixed 
one,  any  point  on  the  radius  of  the  generating  circle  will  describe  a  Limagon, 
and  any  point  on  the  circumference  a  Cardioid. 

Several  interesting  problems  involve  the  four  cusp  hypocycloid  : 
If  the  back  of  a  chimney  be  vertical  and  the  floor  be  horizontal,  and 
the  edge  of  the  front  piece  be  x  feet  from  the  back,  and  y  feet  from  the 
floor  ;  then  the  length  of  the  longest  rod  that  can  be  run  squarely  up  the 

chimney  will  be  the  value  of  R  in  Equation  (10),  that  is,  R  =  (ar '  +  y  ')'\ 
If  a  smooth  bar  rests  on  a  curve  and  against  a  vertical  wall,  the  bar  will 

be  in  equilibrium  in  all  positions  if  the  curve  be  a  certain  hypocycloid. 

The  length  of  the  tangent  of  a  four  cusp  hypocycloid  limited  by  the 

axes  is  constant,  and  equals  the  radius  of  the  directrix. 

293.  The  Lima^on  may  be  deflned  as  in  the  preceding  Remarks.  But 
it  was  originally  defined  as  follows  :  If  a  secant  be  drawn  through  a  fixed 
point  on  a  circle,  and  equal  distances  be  laid  off  both  ways  on  this  secant 
from  the  other  point  where  the  secant  cuts  the  circle,  the  locus  is  a  Lima(;on. 

Taking  the  pole  at  the  fixed  point,  r  the  radius  of  the  circle,  the  initial 
line  passing  through  the  centre,  and  b  the  constant  distance,  then  will  the 
polar  equation  be 

p  =  2r  cos  Q  ±  b, 

and  the  rectangular  equation 

(a;2  +  y«)2  _  (4ra;  +  6')  (ar«  +  y^)  +  4rV  =  0. 

294.  The  Cardioid  is  a  particular  case  of  the  Limai^n,  in  which  6  =  2r. 
The  polar  equation  is 

/o  =  2r(cose  ±  1), 
and  the  rectangular  equation 

(a;2  +  2^8)»  -  4ra!(aj8  +  y^)  -  ^^y»  =  0. 


The  Conchoid  of  Nicomedes. 

295.  If  a  line  OP  be  drawn  tlirougli  a  fixed  point   0 
across  a  fixed  line  CX&i  F,  and  a  constant  distance  FP  be  laid 


298] 


THE  CONCHO W  OF  NIC0MEDE8. 


217 


Fig.  195. 

the  radius  vector ;  tlien  for 


off  on  the  line  both  ways  from  the  point  F,  the  locus  of  the 
point  P  is  called  the  Conchoid 
of  Nicomedes.  The  fixed  point 
O  is  the  iwlc,  XCX  the  direc- 
trix, and  BC  the  parameter  oi 
the  curve, 

296.  To  find  the  equa- 
tion of  the  Conchoid,  let  P 

be  any  point,  BOP=  (p,  OG  — 
a,CB^h=  FP,  and  p  =  OP 
the  polar  equation  we  have, 

OP  --^0F+  FP, 

or  p  —  a  sec  q)  ±  h.  (1) 

For  the  rectangular  equation,  let  x  =  OD,  and  y  =  DP, 
then 

P 

which  substituted  in  (1)  give 

{^  +  y'){y-ay  =  hY.  (2) 

If  the  origin  be  transferred  to  C,  we  will  have  (writing 

y  +  a  for  y) 

x-f^{a+yfih'-f),  (3) 

which  is  the  required  equation. 

These  equations  give 
both  branches  of  the  curve. 
The  branch  nearest  the 
pole  is  called  the  inferior 
branch,  and  the  more  re- 
mote portion  the  superior 
branch. 


sec  (p 


a?  +  y'=  p\ 


Fig.  196. 


If  6  >  a  there  will  be  a  loop  inclosing  the  pole. 


If  a  =  &,  there  will  be  a 
cusp  at  the  pole.  ^ 


218 


HIOEEB  LOCI. 


[297,  298. 


11  h  >  a  the  lower 
brancLi  will  be  more  or 
less  rounded  when  it 
crosses'  the  axis  of  y. 


Fig.  198. 


297.  To  Trisect  an  Angle.— Let  EOC  be  the  angle. 
Take  the  directrix  EG  of  the  Conchoid 
perpendicular  to  OC  at  any  point. 
Construct  a  conchoid  having  its  pole 
at  0,  and  parameter  h  equal  to  ^OE. 
Draw  EP  parallel  to  OC,  P  being 
the  point  where  it  intersects  the  con- 
choid, and  draw  OP  ;  then  will  POC 
=  \EOG.  For,  if  (9  =  EPO  =  POC, 
then  EP  =  b  cos  6  =z  WE  cos  B  ;  and 
the  triangle  EOF  gives 


Via.  199. 


EP  _  sin  (p 
'EO~s^ne 


b  cos^ 


=  2  cos 


or 


.*.  sin  <7?  =  2  sin  d  cos  6^  =  sin  20 

cp  =  2d; 

.-.  d  =  ^cp  =  lEOC, 

which  was  to  be  proved.     Bisecting  EOP  by  well-known 
methods,  the  angle  0  becomes  trisected. 

298.  Remark. — Among  the  noted  problems  of  the  ancient  mathemati- 
cians were  the  Trisection  of  an  Angle  and  the  Duplication  of  the  Cube.  The 
geometrical  construction  of  these  problems  is  beyond  the  reach  of  element- 
ary geometry,  since  it  involves  curves  of  a  higher  order  than  the  circle. 
Both  these  problems  involve  the  solution  of  a  cubic  equation,  and  both  may 
be  made  to  depend  upon  the  construction  of  two  mean  proportionals  be- 
tween two  given  straight  lines.  This  has  been  accomplished  in  various 
ways  by  means  of  higher  plane  curves,  many  of  which  were  invented  for 
this  purpose.  The  Conchoid  was  invented  by  one  Nicomedes,  a  Greek 
mathematician,  for  trisecting  an  angle.  The  Cissoid  of  Diodes  was  in- 
vented by  the  Greek  geometer  Diodes,  for  the  purpose  of  solving  these 
problems. 


299-301.] 


THE  CISSOID. 


219 


The  Cissoid. 

299.  Let  0  be  the  centre  of  a  circle,  AB  a  diameter. 
Erect  a  pair  of  ordinates  DE,  D  E ',  equi- 
distant from  tlie  centre,  and  from  A  draw 
a  secant  AE  through  the  extremity  E 
of  one  ordinate  ;  its  intersection  P '  with 
the  other  ordinate  determines  a  point  in 
the  required  locus.  Similarly  the  inter- 
section of  AE  and  DE  at  P  determines 
another  point.  The  locus  thus  constructed 
is  called  the  Cissoid  of  Diodes. 


300.  Rectangular  Equation  of  the 
Cissoid. — Let  P '  be  any  point  of  the 
curve,  AD '  =  x,  D  P'  =  y,  AO  =  r  ; 
then 

AD  :  DE  ::  AD'  :  DP', 


or  2r  —  X  :  \/{2r—x)x  ::  x  :  y; 

••  ^-  2r-x' 
which  is  the  required  equation. 


Fio.  200. 


(1) 


301.  Polar  Equation  to  the  Cissoid.— Take  the  origin 
at  AyAP'  =  p,P  AX=  e  ;  then 

y  =  p  sin  6,        x  =  p  cos  d, 

which  substituted  in  equation  (1)  will  give 

p  =  2r  sin  6*  tan  6,  (2) 

for  the    required    equation      The  curve  has  two  infinite 
branches  and  is  symmetrical  in  reference  to  the  axis  of  x. 


220 


HIGUER  LOCI. 


[302,  303. 


[Sir  Isaac  Newton  gave  the  following  mechanical  construction  for  this 
curve.  Let  0  be  the  centre  of  the  circle,  AO  =  AD  its 
radius,  and  OC  a  line  perpendicular  to  OA.  TaJce  a 
rectangular  ruler  DEC  whose  leg  EC  equals  OD.  Let 
the  end  C  slide  along  OL  while  EF  constantly  passes 
through  the  point  D,  then  will  P  the  middle  point  of 
EC  describe  a  cissoid. 

The   locus   of  the   vertex   of  a   common  parabola 
rolling  upon  an  equal  parabola  is  a  cissoid.] 


f"^d      a       0 

Fig.  201. 


302.    To  insert  two  mean  proportionals  between  tivo  given 
lines. 

Let  a  and  h  be  tlie  given  quantities.     With  a  =  AC  an  a 
D  radius  describe  a  semi-circumfer- 

ence, and  construct  the  correspond- 
ing cissoid  A  GD.  At  the  centre  of 
the  circle,  erect  an  ordinate  CE  =  h 
and  join  E  and  B,  noting  the  point 
G  where  it  crosses  the  cissoid. 
Draw  GA  and  note  the  point  F 
where  it  crosses  C.fi' ;  then  will  CF 
be  one  of  the  mean  proportionals. 

Let  fall  a  perpendicular  GH 
upon  the  diameter  AB  (not  shown 
in  the  figure),  then  will  x  =  AH,  and  y  :=  GH,  and  the  similar 
triangles  ACF sxidi  AHG,  and  BHG  and  BCE,  give 


a 

CF 

T  2a  —  X 

and ■  = 

_2/ 

X 

y 

a 

6' 

which  combined  with  the  equation  of  the  curve  will  give 

CF^V"^. 

By  exchanging  the  quantities  a  and  h  in  the  construction, 
we  would  find  the  value  of  ^/ab-,  and  the  required  propor- 
tion will  be 

a  :  \/a^b  :  :  \/ab^  :  b. 


303.  Duplication  of  the  Cube. -^T'o /tic?  the  edge  of  a 
cvhe  wlwse  volume  shtll  be  double  that  of  a  given  cube.     Let  a  be 


304,305.]  QUADRATBIX  OF  DIN0STRATU8.  221 

the  length  of  one  edge  of  the  given  cube,  and  making  h  ~  la 
in  the  preceding  Article,  we  have 

which  is  the  required  length. 

To  find  a  cube  whose  volume  is  n  times  that  of  a  given 
cube,  make  h  =na,  and  we  find 


CF  =  aVn 


for  the  length  of  one  edge. 


Quadrat?' ix  of  Dinostratus. 

304.  If  an  ordinate  DP  moves  uniformly  along  the  di- 
ameter of  the  circle  AB,  while  the  ra- 
dius rotates  uniformly  from  B  to  A,       I 
both  beginning  at  the  same  time  at  B,     _[ 
their  intersection  F  will  be  the  locus 
of  the  Quadratrix. 

305.  Equation  of  the   Quadra-  ^.c  m 
trix.— Let  r  =  CB  =  the  radius,  6  =PCB,  x=CD,y=:  DP. 
Then  from  the  definition  we  have 

^^  = ,   y  —  X  tan  6 ; 

r       r  —  X 

.:y  =  xt&ji{r-x)-^, 

which   is  the   equation  of  the  curve.     If  a;  =  0,  ?/  =  0xoo; 
which,  by  the  principles  of  vanishing  fractions,  is  found  to 

be  — ;  hence 

2r 

[Remark. — This  curve  was  invented  by  Dinostratus  for  the  purpose  of 
finding  the  area  of  the  circle  (hence  the  name  Quadi'atrix),  and  also  for 
dividing  an  angle  into  any  number  of  equal  parts.    Thus,  to  trisect  the 


222 


EIOHEB  LOCI. 


[306-308. 


angle  PCD,  trisect  DB  and  erect  ordinates  at  the  points  of  division  ;  then 
•will  the  radial  lines  from  G  to  the  points  of  intersection  of  the  ordinates 
with  the  quadratrix  trisect  the  angle. 

If  the  law  of  construction  for  BEA  be  continued  outside  the  circle  the 
curve  will  become  an  asymptote  to  the  dotted  lines,  and  another  branch, 
shown  by  the  full  lines  outside  the  dotted  ones,  may  be  described,  and  so  on 
indefinitely.]  ' 

Witch  of  Agnisi. 

306.  To  construct  the  Witch  of  Agnisi,  let  AB  be  the  diameter  of  a 

circle  perpendicular  to  AX,  draw  a  tan- 
gent to  the  circle  at  B  (not  shown  in 
the  figure) ;  it  will  be  parallel  to  AX, 
and  through  any  point  F  of  the  circle 
draw  a  secant  AF  and  prolong  it  to 
meet  the  tangent  through  JS  ;  project 
the  intersection  thus  found  on  the  ordi- 
nate EF  prolonged,  the  point  P  thus 
To  find  its  equation  let  x  =  AD,  y=  DP, 


Fig.  204. 
found  wDl  be  a  point  of  the  locus, 
then  we  will  find 


x^y  =  4r^(2r  —  y). 


Ovals. 

307.  An  Oval  is  a  reentrant  curve  in  which  the  distances  of  any  point 
from  two  fixed  points  have  a  constant  relation.  The  fixed  points  are  called 
the  foei.  The  term  is  also  applied  to  figures  made  with  arcs  of  circles, 
which  resemble  the  ellipse  in  form. 

308.  The  Cartesian  Oval  is  one  in  which  a  fixed  multiple  of  one  radius 

vector  of  any  point  differs  from  the 
other  by  a  constant  quantity. 

To  find  its  equation,  let  P  be 
any  point  in  the  locus,  F  and  F'  the 
foci,  p  and  p'  the  radii  vectors,  k  the 
fixed  multiple,  and  d  the  constant 
difference;  then,  from  the  definition, 
we  have 


p  —  kp'=  ±d. 


(1) 


Tig.  305. 
i^'and  the  variable  angle  PFF' 


Let  the  distance  between  tihe 
foci  be  FF'  =  c ;  take  the  pole  at 
g),  then  from  the  figure  we  have 


p'*  =  /3*  +  a^  —  2cp  cos  (p. 


(3) 


.] 


0  VALS. 


223 


Eliminating  p'  gives 

{k-  —  1)  p-  —  2  {ck-  cos  q)Td)p  +  c^A;-  —  d*  =  0,  (3) 

which  is  of  the  form 

p- +2(^ +6cos(p)/3 -^  (7  =  0,  (4) 

and  is  the  required  equation.  Equation  (3)  shows  that  there  are  two  ovals 
answering  the  required  condition,  as  shown  in  the  figure. 

Remarks — This  oval  was  first  investigated  by  Descartes,  hence  its 
name.  It  has  been  shown  that  this  curve  has  a  third  focus  F  outside  of 
both  the  ovals.  (See  Reprint  of  Solutions  from  the  Educational  Times, 
Vol.  XXV. ,  p.  68. )  If  ^  =  —  1  and  d  is  positive,  the  locus  becomes  a  single 
curve  and  is  an  ellipse. 

If  A;  =  1  and  d,  positive,  it  is  an  hyperbola. 
If  rf  =  ck,  we  have,  (Eqs.  (3)  and  (4)), 

p  +  2  ( J.  +  i?  cos  q>)  =  0, 

which  is  the  equation  of  the  lima(;on,  (Art.  293'. 

If  ^•  =  1  and  d  =  c  it  becomes  the  equation  of  the  cardioid,  Art.  294. 

[Mechanical  Construction. — The  following  mode  of  constructing  the  Car- 
tesian is  given  by  Prof.  Hammond,  of  Bath,  England.  A\'iud  a  string 
around  two  concentric  wheels  and  let  it  pass  around 
smooth  pins  A  and  B,  and  be  joined  at  P.  A  pencil 
point  at  P  will  trace  a  Cartesian  when  the  wheels  C 
and  D  are  turned  on  their  common  axis.  To  prove 
this,  differentiate  equation  (1),  and  thus  find  dp  =  kdp  , 
which  gives  the  relation  between  the  rates  of  change  of 
the  radii.  But  from  the  figure  we  see  that  the  rate  of 
increase  of  BP  is  to  the  rate  of  decrease  of  AP,  as  the 
diameter  of  Z),  is  to  the  diameter  of  C.  The  last  ratio 
being  constant,  may  be  represented  by  —  A; ;  which 
shows  that  the  locus  is  a  Cartesian,  the  foci  being  at  A 
and  B.     If  the  circles  C  and  D,  are  equal,  or  ^-  =  —  1,  ^**-  ^^• 

the  locus  is  an  ellipse.  If  the  circles  are  of  the  same  size,  or  both  threads 
are  wound  the  same  way  about  D,  in  which  case  k  =  1,  the  locus  will  be  an 
hyperbola.     {Am.  Jour.  Math.  1878,  p.  283.)] 

Prob. — Prove  that  the  locus  of  the  triple  foci  of  a  series  of  Cartesian 
Ovals  passing  through  five  points  is  an  equilateral  hyperbola.  (Reprint 
Ed.  Times,  Lend.,  Vol.  XVII. ,  p.  24,  1877.) 

309.  The  Cassian  Oval  is  the  locus  of  a  point  the  product  of  whose 
distances  from  two  fixed  points  is  con- 
stant. 

Let  P  be  any  point,  i?'and  i^' the 
fixed  points  called /oci,  0  the  middle 
point  of  FF,x-  OB,  y  =  DP,  OF  =  e, 
TO*  the  constant  product  of  the  radii. 

We  have  from  the  definition  pp  = 
TO*,  and  from  the  figure 

p^  =  (x-e)U  y';  p'«  =(« +  c)*  +  y* 


Fig.  307. 


224 


HIOHEB  LOCI. 


[310,  311. 


These  equations  give 


4c^x^  =:  m*. 


(5) 


which  is  the  rectangular  equation  to  the  curve. 

Changing  to  polar  coordinates,  0  being  the  pole,  r  =  OP  the  radius  vec- 
tor, we  have 

r*  +  2c^  (1-3  cos-  0)r»  =  m'*  -  c*,  (6) 


which  is  the  required  equation. 


310.  liemniscata  of  Bernoulli. — This  is  a  special  case  of  the  Cassian 

in  which  m  =  c,  and  hence  the  rect- 
angular equation  is 

(a;3  +  2/2)^  -2.y>{x^  -y^)  =  0.    (7) 
The  polar  equation  is 

r*  =  2c*  cos  20.  (8) 

If  S  =  0,  we  have  r  ~c  V2,  which 


Fig.  208. 
call  a,  and  the  equation  will  become 


r*  =:a2cos2Q,  (9) 

which  is  the  more  usual  form.     The  curve  crosses  itself  at  the  origin. 

p  If  w  <  c  the  Cassian  does  not  cut 

the  axis  of  y,  and  the  locus  divides  itself 
into  two  distinct  ovals. 

311.  A  Catenary  is  the  curve  as- 
sumed   by  a    perfectly    flexible    chain 
when  suspended  at  its  ends.     If  the  origin  be  at 
the  centre  of  the  curve,  x  horizontal  and  y  verti- 
cal, the  equation  is 


Fig.  210. 


y  =  i^(e^ 


')^ 


in  which  e  is  the  base  of  the  Napierian  system  of  logarithms  and  a  the 
ratio  of  the  weight  per  unit  of  length  to  twice  the  tension  at  the  lowest 
point.    (See  the  Author's  Analyt.  Mech.,  p.  134.) 

[The  following  aro  some  of  the  properties  of  the  catenary: 


to 


Tlie  directrix  is  aline  parallel  to  the  axis  of  x,  and  below  the  vertex  a  distance  equal 
1  . 


If  a  common  parabola  be  rolled  on  a  fixed  line,  thelocnsof  thefocns  will  be  a  catenary;— 
also  the  envi'lope  of  the  directrix  will  be  a  catenary  symmetrical  with  the  former  in  refer- 
ence to  the  fixed  line.    (Reprint,  Ed.  Times,  Vol.  XXV.,  p.  93.) 


312-314.] 


MISCELLANEO  US. 


225 


The  radius  of  curvature  at  any  point  of  tlie  catenary  equals  (in  length)  the  normal 
limited  by  the  directrix. 

The  tension  at  any  point  equals  the  weight  of  the  chain  whose  length  is  the  ordinate 
of  the  point  from  the  directrix. 

If  an  indefinite  number  of  strings  (without  weight)  be  suspended  from  the  catenary  and 
terminated  by  a  horizontal  line,  and  the  catenary  be  then  drawn  out  to  a  horizontal  line,  the 
locus  of  the  lower  ends  of  the  strings  will  be  a  parabola. 

The  centre  of  gravity  of  the  catenary  is  lower  than  for  any  other  curve  of  the  same 
length  terminated  by  the  fixed  points  A  and  Zf.] 


MisceUaneoiis. 

312.  Curves  of  Purstiit. — If  a  point  moves  along  any  patli  and  another 
point  is  made  to  move  directly  towards  it  according  to  any  law,  the  path  of 
the  latter  is  called  a  curve  of  pursuit. 

Problem. -A  fox  runs  uniformly  along  the 
straight  line  AX,  and  when  the  fox  is  at  A  a 
dog  starts  at  C  and  runs  at  an  uniform  rate 
towards  the  fox  ;  required  the  equationof  the  path 
described  by  the  dog,  and  the  distance  ruti  by 
each. 

At  any  instant  let  the  dog  be  at  P  and  the 
fox  at  A',  then  will  PA'  be  a  tangent  to  the 
curve.  This  is  a  curve  in  which  the  length  of 
arc  CP  bears  a  constant  ratio  to  the  distance  AA'. 
Let  AA'  ■+ 
Fluxions), 


A     D 


A'  B 

Fig.  211. 
CP  =  n,  and  CA  -a,x  =  AD,  y  =  DP,  then  it  may  be  found  that  (Simpson'* 


2a;  = 


l  +  n 


ay 


a"  (1  +  '0 


which  is  the  rectangular  equation  to  the  curve.    When  the  dog  overtakes  the  fox,  we  have 

an 
y  =  0;  .-.  x  =  - -7, 


which  will  be  the  distance  run  by  the  fox ;  hence  the  distance  run  by  the  dog  will  be  ^-j^ 

There  are  numerous  curves  ofpurmit  depending  upon  the  laws  to  which  the  moving  bodies 
are  subjected,  and  the  path  described  by  the  leading  body. 

313.  The  Folium  of  Descartes  is  expressed  by  the  equation 


3.3  +  y3  —  Zaxy  =  0. 
It  has  an  asymptote  whose  equation  is 

05  +  y  +  a  =  0. 


314.  Discontinuous  Curves. — The  equations  of 
some  curves  give  real  values  for  certain  values  of  one  of  the  variables  and 
15 


226  EIGHEB  LOCI.  [315-319. 

imaginary  values  for  other  values,  and  when  the  imaginary  parts  fall  be- 
tween  real  portions,  the  locus  is  called  discontinuous. 

315.  The  locus  whose  equation  is 


y  =  ax"^  ±  Vx  sin  hx 

is  an  example  of  a  discontinuous  curve,  in  which  one  portion  of  the  locus  is 
represented  by  points  only.  Thus,  for  negative 
values  of  x,  the  radical  is  imaginary  for  all 
values  of  x  except  when  sin  (—  bx)  is  zero. 
When  that  is  zero  y  is  real  and  gives  the  iso- 
lated points  A',  B',  C,  etc.,  all  of  which  are 
located  on  the  curve  whose  equation  is  i/  =  ax'^. 
All  positive  values  of  x,  (bx  being  less  than  a 
^»-  213-  multiple  of  |vr)  give  real  values  for  y,  and  the 

locus  will  be  a  series  of  ovals   symmetrical  in  reference  to  the  positive 

branch  of  the  parabola  y  =  ax'^ . 

316.  The  locus  y  =  l/3  sin  a;  —  1  •  Vcos  x  —  1,  is  another  example. 
If  a;  >  0  and  <  80°  both  radicals  are  imaginary  and  hence  y  will  be  real. 
For  X  >  30^  and  <150',  the  first  is  real  and  the  second  imaginary,  and  hence 
y,  imaginary  ;  and  so  on  throughout  the  circumference  and  multiples  of  the 
circumference. 

317.  If  the  locus  be  referred  to  polar  coordinates,  the  same  condition 
may  exist,  as  may  be  seen  from  the  equation 

p=  V2  -  3  sin  49>. 

318.  A  Loxodromic  Curve  is  one  that  cuts  the  meridians  of  a  sphere 
at  a  constant  angle.  It  is  found,  by  higher  analysis,  that  the  equation  of 
a  loxodromic  of  45°  is 

X  =  log  tan  (45°  +  \y), 

i^i  which  y  is  the  latitude  and  x  the  longitude,  the  origin  being  on  the 
equator. 

[Remark.— If  a  ship  should  start  at  the  equator  and  sail  continually  nsrth-east  at  a  finite 
rate,  it  would  reach  the  north  pole  in  a  finite  time.  It  would  go  around  the  pole  an  infinite 
number  of  times,  but  the  length  of  the  path  would  be  finite.  When  it  passed  360  degrees  of 
longitude,  its  latitude  would  be  89°  53',  or  it  would  be  within  about  8  miles  of  the  pole.] 

319.  The  Logocyclic  Curve  is  one  whose  polar  equation  is 

p  =  a  sec  0  (1  ±  sin  6), 

in  which  a  is  the  value  of  p  f or  6  =  0.     The  rectangular  equation  is 

(ajs  +  y^)  {x  —  2a)  +  a^x  -  0. 

The  locus  of  the  foci  of  all  elliptic  sections  whose  planes  pass  through 
a  tangent  to  a  circular  cylinder  parallel  to  the  base  is  a  logocyclic  curve. 


320-323.] 


MISCELLANEO  US. 


227 


320.  An  Helix  is  a  curve  wliich  cuts  tlie  rectilinear  ele- 
ments of  a  cylinder  at  a  constant  angle. 

To  find  its  equations,  let  the  axis  of  z  coincide  witli  tlie 
axis  of  the  cylinder,  x  and  y,  horizontal,  6,  the  variable  angle 
measured  from  the  axis  of  x,  r,  the  radius  of  the  base,  and 
(p  the  angle  between  the  helix  and  the  rectilinear  elements ; 
then  we  find 


x  =  r  cos 


y  =  r  sin  6,        z  =  r  6  cot  cp, 


which  are  the  required  equations.  If  c  be  the  slope  of  the 
helix  ;  that  is,  the  tangent  of  the  angle  which  the  helix  at  a 
unit's  distance  from  the  axis  makes  with  the  base,  then 

z  =  cd, 

and  the  values  of  x  and  y  remain  the  same. 

321.  A  Conoid  is  a  surface  generated  by  a  right  line 
remaining  constantly  parallel  to  a  plane  and  mo\dng  on  two 
other  lines  one  or  both  of  which  is  curved.  The  plane  is 
called  a,  plane-directer.     It  is  a  warped  surface. 

322.  Problem. — To  find  the  equation  to  a  conoid  in  which 
owe  directrix  is  an  ellipse  and  the  other  a  right  line,  the  right  line 
being  parallel  to  the  major  axis  of  the  ellipse  and  perpendicular  to 
the  plane-directer. 

Let  the  axis  of  x 
coincide  with  the  ma- 
jor axis  of  the  ellipse, 
and  the  plane  yz  be  the 
plane-directer  passing 
through  the  principal 
vertex  of  the  ellipse; 
BE  the  generating  line, 
P,  any  point  in  the  sur- 
face, h  =  CB,  the  alti- 
tude ;  x  =  AC,  y  =  CD, 
and  z  =  DP.  The  equa- 
tion to  the  ellipse  gives, 
(Art.  72), 


Pig.  214. 


a* 


228  HIGHER  LOCI.  [8221 

and  the  similar  triangles  EDP  and  ECB  give 

ED  :  EC  .:  DP  :  CB, 
or  EC-y  \  EG  \\z  \h\ 

hence        '  hy  =  {h  -  zf  EC^  •, 

.:  a^hY  =  h\}i  -  zf  {2ax  -  a:?), 
which  is  the  required  equation. 

EXAMPLES. 

1.  Find  the  curve  of  intersection  of  a  plane  with  the  conoid. 

2.  Show  that  a  plane  section  parallel  to  the  curved  directrix  is  a 
carve  of  the  same  class. 


PART   II. 


QUATERNIONS. 


2^ 


{V=^' 


QUATERNIONS. 


CHAPTER  I. 


ADDITION     AND    SUBTRACTION    OF    VECTORS. 
Definitions. 

323.  Quaternions  is  a  system  of  analytical  geometry, 
invented  by  Sir  William  Rowan  Hamilton  about  the  year 
1843.  He  gave  to  the  system  the  above  name  because  it  in- 
volves, in  its  fundamental  expressions,  four  arbitrary  units 
(from  the  Latin  word  quaternio,  meaning  a  set  of  four.)* 

324.  A  Vector  implies  the  transferrence  of  a  point  a 
given  distance  in  a  given  direction.  Thus,  if  a  point  be  trans- 
ferred from  A  to  B,  the  length   and  direction  of  AB  being 

known,    any   quantity   which    will   rep-     a  — ■ b 

resent  this  action   is  a  vector.     There-  °  ° 

fore,  in  this  svstem,  a  vector  is  the  repre-  ^ 

^  ~  ^  .  flG.  215. 

sentative  of  transferrence  through  a  given 
distance  in  a  given  direction.  Geometrically  it  is  represented 
by  a  right  line  whose  direction  is  parallel  to  the  transferrence, 
and  whose  length  equals  the  distance  through  which  the 
point  has  been  carried.  Analytically  it  is  represented  by 
some  letter  of  the  Greek  alphabet,  a,  0,  y,  etc. 

325.  The  Sign  of  a  Vector. — If  the  transferrence  be 
considered  as  positive  in  one  direction,  a  transferrence  in  the 
opposite  direction  will  be  negative.  Either  direction  may 
be  assumed,  arbitrarily,  as  positive.     Thus,  ii  AB  be  posi- 


*  Hamilton's  Lectures,  Preface,  pp.  (46),  (62) ;  also  pp.  89,  109,  112,  128, 
449. 

231 


232  QUATERNIONS.  [326,327. 

tive  BA  will  be  negative,  the  letters  being  arranged  in  the 
order  of  the  transferrence,  AB  signifying  a  transferrence 
from  A  towards  B.  This  principle  will  be  observed  in  this 
system. 

326.  Equal  Vectors  are  such  as  are  parallel  and  equal 

in  length.     Thus   if  AB,   CD, 
Cap  and  EF'  are  parallel  and  equal 

tK    CL     la         /  ill  length,  we  write 

H^  3ff^ 7^ 

ir-ar-i'  AB=CD=EF. 

^'"'  ^^^'  Equal  vectors  are  added  the 

same  as  similar  quantities  in  algebra ;  hence  we  have,  for 
this  case, 

AB+  CD  +  EF=SAB. 

If  HG  is  parallel  to  AB  and  equal  in  length  to  SAB,  we 
have 

AB+  CD  +  EF=  EG  =  3J5, 

or  AB+  CD  +  EF+  GH=  0. 

327.  Parallel  Vectors  are  Multiples  of  each  other. 

This  follows  directly  from  the  preceding  Article.  Since 
they  are  parallel,  we  have,  in  comparing  one  vector  with 
another,  only  to  compare  their  lengths.  If  a-  be  a  vector, 
then  will  na  be  a  parallel  vector  n  times  as  long.  In  the  tri- 
angle CDF,  if  DF  is  parallel  to  AB 
and  n  times  as  long,  and  if  vector 
AB  be  a,  then  will  vector  DF  be  ncv. 
Generally,  we  have 

a  +  la-^  ma  +  etc.  =  {l  +  l  +  m  +  etc.)  a. 

in  which  I,  m,  etc.,  may  be  positive 
or  negative,  entire  or  fractional. 
Similarly,  in  Fig.  216,  H  HA  = 
§,  BC=l/3,  DE=  -  n0,  etc.,  we  have 

ft  -^  Ifi  —  nft  +  etc.  =  (1  +  ?  -  w  +  etc.)/?. 

Vectors  not  parallel  must  not  be  represented  by  the  same 
letter. 


Fig.  217. 


828-330. J  '    VECTOR  EQUATIONS.  233 

328.  A  Unit  Vector  is  one  wliose  length  is  unity,  its  di- 
rection being  given  or  assumed.  The  length  of  the  unit  will 
be  the  same  for.  all  the  vectors  in  any  particular  problem. 
Unit  vectors  are  generally  represented  by  the  same  Greek 
letters  as  the  entire  vector,  in  which  case  they  are  especially 
designated  as  such ;  thus,  let  a,  fj,  y,  etc.,  be  unit  vectors, 
but  we  will,  at  present,  distinguish  them  by  subscripts,  thus, 
^\,  Aj  Yu  etc.,  are  unit  vectors,  of  which  n,  fi,  y,  etc.,  are  the 
entire  corresponding  vectors,  and  we  may  have 

a  =  la'i,  (5  =  xftx,  etc., 

and  similarly  for  the  others. 

329.  A  Tensor*  is  the  numerical  factor  by  which  a  unit 
vector  is  multiplied  to  produce  the  real  vector.  Thus,  I,  m, 
n,  etc.,  in  the  preceding  expressions,  are  tensors.  Tensors 
were  represented  by  Hamilton,  in  general  discussions,  by  the 
letter  T;  thus,  Ta,  Tft,  etc.,  and  are  read,  'tensor  <■;,  tensor 
/?,'  etc.  This  notation  is  still  retained  in  many  cases. 
Thus,  we  have 

a^Ta  (a,),         fd  ^  T /3  (fJ,),  etc. 

(Remakk. — The  definition  of  the  other  terms.  Scalar  and  Versor, 
will  be  given  in  the  second  chapter.  A  Quaternion,  strictly  speaking,  in- 
volves all  four  units,  and  hence  the  analysis  of  this  chapter,  involving,  as 
it  does,  only  two  of  the  required  units,  is  at  best  a  restricted  and  partial 
case  of  the  more  general  analysis.  Still,  the  principles  here  developed  are 
a  necessary  part  of  the  subject.) 

Vector  Equations. 

330.  Let  ABC  be  a  triangle,  the  direction  and  length  of 
AB  being  represented  by  vector  a, 
of  BC,hj  vector  p,  of  AC,  by  vec- 
tor y.  The  transferrence  of  a  point 
from  A  to  B,  followed  by  a  trans- 
ferrence from  JS  to  C,  gives  the 
same  result  as  a  transferrence  di- 
rectly from  A  to  C.  This  is  expressed  in  the  form  of  an 
equation,  thus 

a  +  /3  =  r, (1)^ 

*  Literally,  that  which  stretches. 


234  QUATERNIONS. 

The  symbols  +  and  =  have  not  the  same  meaning  here 
as  in  algebra.  The  addition  is  not  numerical,  neither  is  the 
equality  a  numerical  one ;  still,  their  meaning  here  is  not  op- 
posed to  that  in  algebra.  They  are  used  in  an  enlarged  sense. 
The  expression  may  be  read  '  a  traneferrence  expressed  by 
vector  (X,  followed  by  a  transferrence  expressed  by  vector  ^, 
is  equivalent  to  a  transferrence  expressed  by  vector  y.'  In 
this  sense  the  expression  may  be  called  a  vector-equation,  and 
read  in  the  usual  way,  thus, '  a  plus  fS  equals  y.'  If  the  vec- 
tors are  parallel,  they  will  be  represented  by  multiples  of 
the  same  vector,  and  the  equation  will  express  a  numerical 
equality,  (326). 

331.  Law  of  Signs. — A  separate  vector  may  be  posi- 
tive in  either  direction  along 
the  line,  (325).  When  they  are 
connected  with  each  other  the 
sign  depends  upon  the  order  of 
the  transferrence.  Thus,  if  the 
transferrence  be  from  A  to  B, 
thence  from  B  to  C,  thence  from  C  to  A,  and  all  these  direc- 
tions be  considered  positive,  we  have 

a+  ft  +  y=Q.  (2) 

But  if  we  make  AB,  BC,  positive,  and  also  AC  positive 
from  A,  then  will  CA  be  negative  ;  in  which  case  we  have 

a  +  ft-y  =  0,  (3) 

which  is  the  same  as  transposing  y  in  equation  (1)  to  the 
first  member ;  hence,  The  rule  for  the  transposition  of  terms  in 
a  vector  equation  in  regard  to  signs,  is  the  same  as  for  algebraic 
equxjbtions. 

If  the  transferrence  be  positive,  and  from  B  to  A,  and  A 
to  O,  the  result  will  be  the  same  as  from  B  to  C,  and  we 
have 

a  + y  =  ft. 

Or,  if  the  transferrence  had  continued  from  0  towards  B, 
BC  being  positive,  we  would  have 

a  +  y-ft  =  0. 


332,  333.] 


VECTOR  EQUATIONS. 


235 


B 


The  transferrence  may  begin  at  any  angle  of  the  triangle. 

[Obs. — This  freedom  in  regard  to  signs  may,  at  first,  seem  to  lead  to 
uncertainty  in  regard  to  the  result  ;  but  it  is  only  necessary  to  observe  that 
the  result  must  be  interpreted  in  accordance  with  the  original  assumptions. 

If  the  right  member  consists  of  one  term  only,  it  may  be  considered  as 
a  measure  of  the  result,  while  the  left  member  may  be  considered  as  the  ex- 
pression of  an  operatio7i.] 

332.  Unless  otherwise  stated  the  vectors /row  the  initial 
point  will  be  considered  positive ;  but  tJie  directions  may  he 
assumed  arbitrarily. 

EXAMPLES. 

If  vectors  AB  =  a,  BC  =  ft,  CD  =  y, 
AD  =  S,  excepting  that  the  positive  signs 
are  not  necessarily  in  the  order  of  the  let- 
ters here  given;  interpret  the  following 
vector  equations;  tliat  is,  give  the  order  of 
the  transferrence,  the  position  of  the  point 
after  transferrence,  and  the  direction  in 
which  the  vector  is  positive.  "  H 

Fig. 220 
a  +  /3  =  r  +  S 

a  =z  y  +  d  —  /3 
a  +  /3  +  y  +  S  =0 

0  =  d  +  y  +  /3+a 
a  +  /3  +  y  =d 
AH+HO  =  AO 

AO  =  AD  +  DG-  OG. 

333.  The  sign  of  a  vector  in  one  direction  being  fixed, 
the  other  vectors  paral- 
lel and  in  the  same  di- 
rection should  have  the 
same  sign.  Thus  if  BD  be 
positive  from  B  towards 
D ;  then  should  BE  and 
EA  also  be  positive.  This 
principle,  if  observed,  will 
prevent  confusion  in  re- 
gard to  signs. 


ma 


I 

Fia.  221. 


236  QUATERNIONS.  [3Si 

Exercise. — Let  a^^  /?, ,  y^  be  unit  vectors,  and  mo'i,  n^iy 
lyi,  the  sides  of  the  triangle.  Draw  radial  vectors  CD,  CE, 
from  C ;  and  let  x  =  BD,  y  =  DE,  z  —  EA ;  it  is  required  to 
show  that  lyx  —  ma^  +  %A  by  passing  from  the  triangle  BCD 
to  DGE,  and  thence  to  EC  A. 

We  have,  (330), 

for  triangle  BCD     xy^  =  ma^  +  CD, 
DEC    yy,  =  -  CD  +  CE, 

EC  A     Byi=-CE  +  nj3i. 

Adding  we  have 

{x  +  y  +  z)yi  =  moci  +  n^i . 

But  X  +  y  +  z  =:l; 

.:  lyi=  moi-y  +  w/?i, 
as  required. 

Let  the  reader  deduce  the  same  result  by  taking  D  as  the 
initial  point. 

334.  Proposition.— If  2a  +  :2ft  =  0,  then  will  2a  -  0, 
and  2ft  =  0. 

For  the  vectors  being  entirely  independent  of  each  other, 
either  sum  may  be  zero  independently  of  the  other. 

We  may  also  show  it  directly  from  Article  326 ;  thus,  no 
amount  of  transferrence  along  vectors  a  will  affect  ft,  hence 
if  their  sum  is  zero,  each  must  separately  be  zero.  This 
may  be  illustrated  by  the  figure.     We  have 


AB  +  CD+  EE+  GH=0=  2a, 


G 


BA  +  BC+  DE+  FG  =  0=^  2ft, 
Fig.  222.  and  2a  +  2ft  =  0, 

from  which  we  see  that  each  may  be  independently  equal  to 
zero.     The  expressions  developed  become 

(l  +  m  +  n+p  +  etc.)a  =  0, 

(a+b  +  c  +  d+  etc.)/?  =  0. 


334.] 


VECTOR  EQUATIONS. 


237 


This  principle  is  similar  to  that  of  indeterminate  coeffi- 
cients in  Algebra. 


APPLICATIONS. 

1.  The  corresponding  sides  of  mutually  equiangular  triangles 
are  proportional. 

Let   ft"! ,  A )  Ti ,   be   unit   vectors  ;  C 

vector  BC  —  ma^,  vector  CA  —  n/^^ , 
vector  BA  —  lyi.  (Observe  that  a  is 
opposite  A,  ft,  opposite  B,  and  r,  op- 
posite (7,  corresponding  to  similar 
notation  in  trigonometry.)  Then, 
since  the  sides  of  the  triangle  CDE 
are  parallel  to  ABC,  we  will  have 

Fig.  223. 

EC  =  aa, ,         CD  =  6A ,        ED  =  cy^ . 
From  the  triangle  ABC  we  have,  (330), 

wo-i  +  nfti  =  lyi ,  (1) 

and  from  CDE  aa^  +  bfti  =  c^j.  (2) 

Multiplying  the  first  equation  by  c ;  the  second  by  I ;  and 
taking  the  difference,  gives 

(cm  —  la)ai  -r  (en  —  lb)fti  =  0. 
Hence,  according  to  Article  334,  we  have 
cm  —  la  =  0, 
en  —  lb  =0; 
cm  _  la 
' '   en       lb' 

and  dropping  the  common  factors  c  and  I,  and  multiplying 
by  ai  and  dividing  by  fti  gives 

nfti      bfti ' 
and,  substituting  lines  for  the  corresponding  vectors,  we  have 


CB 
CA 


CE 
CD' 


238  QUATERNIONS.  [384. 

or  CB  :  CA  ::  CE  :  CD, 

which  was  to  be  proved. 

Similarly,  eliminating  a^  from  equations  (1)  and  (2)  will 
give 

CA  :  AB  ::  CD  .BE. 

2.  If  the  opposite  sides  of  a  quadrilateral  are  parallel,  they 
equal  each  other  in  length. 

The  term  vector  being  understood,  it  is  generally  omitted 

a a     in  speaking  of  a  line.     Thus,  instead 

Z'^"    ~~^^^^/        of  saying  'vector  CD  =  o','  we  simply 
y  ^.><r     /         say  '  let  CD  =  a.' 
P^^:— -^— — ^  Let  CD  =  o',  CA  =  /3;  then  since 

Fig.  224.  AB  is  parallel  to  CD  it  will  equal  ma; 

and  similarly  DB  =  n^. 

Calling  CD  positive,  BA  will  be  negative.     We  have 
vector  CD  +  vector  DB  +  vector  BA  +  vector  AC  =0, 

or  a  +  n/3  —  ma  —  /?  =  0 ; 

or  {l-m)a-  {l-n)^=0, 

which,  according  to  Article  334,  gives 

1  —  m  =  0,        1  —  n  =  0,; 
.'.  m  =  l,  and  n  =  1; 
therefore  AB  =  CD,        CA  =  DB, ' 

which  was  to  be  proved. 

8.  If  the  lengths  of  the  opposite  sides  of  a  qvadrHoierol  are 
equal,  they  uM  be  'parallel. 

^0  Let  the  unit  vectors  along  CA,  CD, 

/~\^  ^_^^^^^l      DB,  AB,  be  A  j  ^i  >  Yx  >  ^\  >  respective- 
'         ^^*^       '  ^      ly ;  then  will  the  sides  as  vectors  be 


334.]  VECTOR  EQUATIONS.  239 

and  we  have,  (330), 

mcx-^  +  nyi  —  md^  —  nl3^  =  0, 
or  («'i  —  Si}m  +  {ri  —  /3i)n  =  0. 

But  since  m  and  w  are  mutually  independent,  we  have 

«'i  -  ^1 ,        n  =  Pi  y 

hence  the  opposite  sides  are  parallel  and  the  figure  is  a  paral- 
lelogram. 

4.   The   diagonals 
of    a    parallelogram 
mutimUy    bisect    each 
other. 
Let 

AB=  DG=of, 

AD=BC  =  ft; 

a  unit  vector  along  AE  =  6^ ,  and  along  BD  =  yi .     Then, 
(327), 

AE=xS,,       EC  =  ydi, 
BE=uyx,       ED  =  vyi. 

We  have,  (330), 

AE=AB  +  BF, 

or  x^i  =  a  +  uyx ; 

also  ydx  ■=  vy-i  +  a. 

Subtracting,        (x  —  y)^^  =  (u  —  v)yx , 
or  {x-y)Sx-{u-v)yx  =  0; 

.'.     (334),  x=y,  u  =  v, 

or  AE=EC,         BE=EDy 

which  was  to  be  proved. 


240 


QUATERNIONS. 


[884. 


5.  The  lines  joinmg  the  middle  points  of  the  opposite  sides  of 
ANY  quadrilateral  mutually  bisect  each  other. 

The  proposition  will  be  proved 
to  be  true  whether  the  lines  all  lie 
in  one  plane  or  not.  If  they  do  not, 
the  surface  will  be  warped,  (222), 
and  is  sometimes  called  a  surface 
gauche,  a  French  word  signifying 
twisted. 

Let  three  vectors,  AB,  AC,  AD, 
co-initial  at  A,  be  drawn  in  any  direction  and  of  any  length, 
and  their  extremities  joined  by  the  lines  B  C,  CD,  making  a 
quadrilateral  A  BCD.  (Vector  AG  \^  not  shown  in  the  fig- 
ure.) The  three  vectors  here  given  will  always  determine 
the  figure,  and  by  using  them,  instead  of  the  four  sides  as 
vectors,  the  solution  is  simplified,  and  a  more  symmetrical 
expression  found  for  the  result. 

Let  AB  —  a,  AC=  /3,  AD  =  ;/,  all  positive  from  A  ;  E, 
G,  F,  H,  middle  points  of  the  respective  sides,  and  0  the 
middle  point  of  GH',  we  are  to  prove  that  it  is  also  the 
middle  point  of  EF. 


We  have 


or 

and 

also, 

or 


and 


then. 


AD  +  DC  =  AC, 

y  +  DC^ft', 

.'.DC=p-y, 

DG  =  l{l3-y); 

AH+  HG  =  AD^  DG, 

ha  +  HG=y  +  U/3-r) 

=  hi^  +  r); 

.-.   HG  =  ^{/3  +  y-a), 

EO=^HG 

=  h{ft  +  y-a); 

AO  =  AH+  HO 

=  la  +  \{fi  +  y- 

a) 

^l{a  +  ^  +  y). 

(1) 

835.]  VECTOR  EQUATIONS.  241 

Let  the  middle  point  of  EF  be  0\  tlien 

AO'  ^AE+EO' 

=  hy  +  lEF.  (2) 

To  find  EF,  we  have 

AE+EF  =  AB  -F  BF; 
.-.  EF^a  +  lBC-^r, 
and  ^EF=^a-^^y+\BC.  (3) 

To  find  BC,  we  have 

^^  +BC=AC, 
or  ^C=  /5  -  «, 

and  '^BC='^{/3-a),  (4) 

which  in  (3)  gives 

and  this  in  (2)  gives 

A0'=\{a  +  /3+r),  (5) 

which  being  identical  with  equation  (1)  shows  that  the  points 
O  and  0  coincide  ;  hence  the  lines  EF  and  HG  mutually 
bisect.  The  point  0  is  called  the  mean  point  of  the  poly- 
gon- 
It  will  be  observed  that  the  mode  of  solution  consists, 
chiefly,  in  reaching  the  same  point  by  two  different  routes. 

335.  The  mean  point  of  a  polygon  is  a  point  to  which 
the  vector  is  the  average  of  the  vectors  to  all  the  angles 
of  the  polygon.  The  initial  point  may  be  chosen  arbitra- 
rily. 

The  mean  point  coincides  with  the  centre  of  gravity  of  a 
system  of  equal  particles,  one  particle  being  at  each  angle 
of  the  polygon. 
16 


242 


QUATERNIONS. 


[335. 


6.   Tlw  mean  point  of  a  quadrilateral  is  at  the  middle  point 
0  c       9f  ^^'6  line  joining  the  points  of  bisection  of 

th£  diagonals. 

Using  the  notation  of  the  preced- 
ing example,  and  letting  0  be  the  mid- 
dle point  of  PQ,  we  have, 


Fie.  238. 


AO  =  AP  +  PO 

=  lfi  +  l{AB  +  BQ-AP)* 

=  \ft  +  la-{.\BD 

=  \/3+}^a+l{AD-AB) 

=  h{oc+  ft  +  y)', 

hence  the  point  0  of  this  figure  coincides  with  0  of  the  pre- 
ceding figure. 

An  interpretation  of  this  result  gives  a  geometrical 
method  of  finding  the  mean  point,  for  A:A  0  is  one  side  of  a 
polygon  of  which  the  other  sides  are  a,  /?,  y. 


*  A  little  practice  will  enable  the  student  to  make  these  substitutions 
Vrithout  formally  writing  the  equation,  and  determining  the  value  of  the 
unknown  quantity.     In  this  case  the  equation  would  be 

AP+  PQ=AB  +  Bqi 
.'.  PQ  =  AB  +  BQ  -AP, 

which  compared  with  the  figure  shows  that  the  terms  to  be  substituted  will 
be  the  remaining  sides  of  the  polygon  APQB,  and  that  the  signs  of  the  vec- 
tors will  be  determined  by  passing  backwards  around  the  polygon.  Thus, 
instead  of  passing  around  in  the  direction  from  P  towards  Q,  go  in  the  oppo- 
site direction.  As  AP  was  considered  positive  from  A  towards  P,  it  will 
be  negative  from  P  towards  A,  and  we  have  —  AP.  Similarly,  +  AB  and 
+  BQ.  The  order  of  the  sides  (or  terms)  is  immaterial,  thus  we  may  have 
-f  BQ  —  AP  +  AB.  It  would  be  well  for  the  student  to  mark  on  the 
figure  the  positive  directions  of  the  vectors,  as  soon  as  the  direction  has 
been  fixed,  as  shown  in  Fig.  832. 


885.] 


VECTOR  EQUATIONS. 


243 


Construct  a  polygon  whose  successive  sides  are  a-,  ft,  y, 
and  join  AD'.     Then 
we  have,  (330), 

AI)'=  a  +ft  -hy, 

and 

AO  =  k(^  +  ^  +  r)> 

.'.  AO  =  \AD. 

The  order  of  the 
sides,   as    shown  by  '^ 
the  equation,  is  im- 
material, hence  the  successive  sides  may  be  «",  y,  ft,  or  y,  a, 
ft.     It  is  not  necessary  for  them  to  be  all  in  one  plane. 

To  find  the  mean  point  we  simply  add  the  vectors  and 
divide  by  the  number  of  angles  of  the  polygon. 


EXAMPLES. 

1.  Prove  that  the  mean  point  of  a  triangle  is  in  a  line  joining  the 
vertex  with  the  centre  of  the  base,  and  at  two-thirds  its  length  from  the 
vertex.  (It  will  be  at  the  same  point  as  the  centre  of  gravity  of  three 
equal  particles,  one  at  each  angle  of  the  triangle.) 

2.  Take  the  initial  point  on  the  diagonal  of  a  square  prolonged,  and 
show  that  the  mean  point  of  the  square  is  at  the  middle  point  of  the  di- 
agonal of  the  square. 

(Obs. — Let  the  adjacent  sides  of  the  square  be  ex,  fi,  and  the  vectors  to 
the  four  angles  be  y,  8,  e,  u,  then  if  S  be  the  vector  to  the  nearest  angle, 
prove  that  AO  (the  mean  vector)  equals  8  -f-  ^(a  +  fi.) 

3.  If  the  initial  point  be  at  the  middle  of  the  base  of  a  triangle, 
prove  that  the  vector  to  the  mean  point  is  one-third  the  vector  to  the 
opposite  angle. 

(Obs. — These  examples  should  be  solved  by  writing  the  equations,  and 
not  deduced  from  the  value  of  ^0  given  above.) 

4.  In  a  regular  pyramid  having  a  rectangle  for  the  base,  equal  heavy 
particles  are  placed  one  at  each  comer  of  the  base  and  vertex  of  the  pyra- 
mid ;  show  that  the  centre  of  gravity  of  the  five  particles  will  be  on  the 
altitude,  and  at  ^  its  length  from  the  vertex. 


244 


QUATERNIONS. 


[836,  337. 


Medial  Vectors. 

336.  A  medial  vector  is  one  drawn  from  the  common 
point  of  two  given  vectors  to  the 
middle  of  the  line  joining  the  ex- 
tremities of  the  given  vectors. 

Thus  if  OA  and  OC  are  given 
vectors,  and  OB  a.  line  drawn  from 
0  to  B,  the  middle  of  AC,  then  will 
OB  he  the  medial  vector. 

337.  Problem. — To  find  an  ex- 
pression for  a  medial  vector. 

Let  a  and  y  be  the  given  vectors, 
and  fi  the  medial,  all  positive  from 
the  initial  point.     We  have 


Fig.  230. 


OB=OA  +  AB 
or  /3  =  a+ABi 

also  OB  +  BC=  OC, 

or      ^  /3  =  r-BC; 

adding  gives  ^  =  h  (^  +  y)>  since  AB  =  BC,   (1) 

which  is  the  required  value. 

The  rules  for  signs  apply  to  this  expression  the  same  as 
for  other  cases.    If  y  be  positive  towards  the  origin,  we  have 


^=H«-r); 


(2) 


and  if  a  and  y  are  both  positive  towards  the  origin,  and  /? 
positive /rom  it,  we  have 

^=-i(a+y), 
or  -^  =  ^(a  +  y);  (3) 

and  if  a  and  y  are  both  positive /rom  the  origin  and  /3  posi- 
tive towards  it,  we  have 

or  ^=:-i(^a+y),  (4) 

which  are  the  same  as  the  two  preceding  equations. 


337.] 


MEDIAL    VECTORS. 


245 


APPLICATIONS. 

1.   One-half  the  diagonal  of  a  parallelogram  wJiose  adjacent 
sides  are  a   and  fS  is 
the     medial    of    those 
sides. 

In  tlie  parallelo- 
gram A  BCD  we  are 
to  show  that  one- 
half  of  A  C  is  the  me- 
dial of  AB  and  AD. 

Let  AD  ^  13  = 
EC,  and  AB  =  a;  then 


Fig.  231. 


AC=a+  13', 
.:AE=^AC, 

hence,  according  to  equation  (1),  AE  is  the  medial  oi  AB 
and  AD,  and  E  is  in  the  line  JjD.  Similarly,  BE  =  ^  ED  is 
the  medial  of  BA  and  EC.  This  is  another  mode  of  prov- 
ing that  the  diagonals  of  a  parallelogram  are  mutually  bi- 
sected. 

2.   The  medials  of  a  triangle  meet  in  a  point  and  mutually  tri- 
sect each  other. 

ft 

(The  medials  are  the  lines  from 
the  angles  to  the  middle  of  the  oppo- 
site sides.) 

Let  AM,  BN,  CL,  be  drawn 
from  the  angles  to  the  middle 
of  the  opposite  sides,  and  take 
the  vectors  all  positive  in  a 
left-handed  direction  starting 
from  A,  in  the  triangles  ^/>* (7,  AOC,  AON,  as  indicated  by 
the  arrow-heads.  The  lines  AM  and  EN  will  intersect 
at  some  point,  as  0;  draw  from  0  the  line  OC,  then  if  OC 


Fig.  232. 


246  .  qUATEBNIONS.  [837. 

and  the  medial  LC  coincide,  the  three  medials  will  all  meet 
in  0. 

'LetAB  =  y,  BC  =  a,  CA  =  /3,  AO  =  d,  0M=m6,  B0  = 

We  have  AB  +  BG+  GA^O, 

or  y  +  a  +  /3  =  0;                  (1) 

also,  AB  +  BO-AO  =  0, 

or  y  +  e-S=0;                  (2) 
also,                     OM+MC+  CN+  N0  =  0, 

or  7nS+  ^a  +  ^/3-n£  =  0.                   (3) 

From  (1)  and  (2),  d  +  a  +  /3-€  =  0;                  (4/ 

from  twice  (3)   subtract  (4), 

{2m-l)S-{2n-l)€  =  0; 
therefore,  (334),  2m  =  1,  and  2w  =  1 , 
or  A0  =  20M  and  BO  =  20N, 

or  AM=  303/ and  BN=  SON; 

hence  A3f  and  BN  trisect  each  other.     (Equation  (4)  may  "be 
deduced  directly  from  the  quadrilateral  AOBCA.) 

Next  we  have 

A0+  OC+CA  =  0, 
or  6  +  00+  /3  =  0; 

.'.  00= -/3-d 

=-^-^AM 
because  AM  is  medial,  =— ^  —  ^xh{y~^) 

from(l),  =-/3-^{-a-2^) 

But  because  CL  is  medial,  we  have 
-LC=hi/3-a\ 


338.]  CO-PLANAR    VECTORS.  247 

or  LC=l{a-§):, 

.'.  OG=ILC, 

therefore,  (327),  CL  and  CO  are  parallel,  and  because  they 
have  the  point  C  common  they  coincide ;  and  the  medial  CL 
passes  through  0 ;  and  since  CO  =  |  CL  it  is  trisected  at  0. 

(Obs. — In  order  to  find  the  numerical  values  of  the  tensors  we  find  an 
equation  between  two  vectors  and  apply  Article  334.) 

EXAMPLES. 

1.  If  a,  /3,  }^,  make  a  closed  triangle,  and  5  is  medial  between  a  and 
y,  find  its  value  in  terms  of  a  and  /5. 

2.  If  fd  =  ^  (a  +  y)  is  the  medial  of  a  and  y,  all  being  positive  from 
the  origin;  if  the  vectors  be  prolonged,  tlie  negative  vectors  being 
equal  to  the  positive  ones,  and  the  medial  positive  from  the  origin; 
show  that  the  medial  in  one  of  the  supplementary  angles  will  be  fi'  — 
I  {a  —y)i  and  in  the  other  fi"  =\  {—a  +y). 

3.  Of  three  co-initial  vectors  a,  /?,  y,  find  the  medial  of  the  medials 
of  a,  ^,  and  fi,  y. 

4.  Given  three  co-initial  vectors  a.,  ^,  y\  if  the  extremity  of  the 
medial  of  a  and  /3  be  joined  by  a  straight  line  to  the  extremity  of  the 
medial  of  (S  and  y ;  show  that  it  will  be  parallel  to  the  line  joining  the 
extremities  of  a  and  y. 

Co-planar  Vectors. 

338.  Any  three  co-planar  vectors  which  give  tie 

relation 

aoi  -\-hfi  ->r  cy  =  0, 

may  he  represented  by  the  sides  of  a  triangle. 

For  a  line  may  be  drawn  equal  and  parallel  to  aor,  and 
through  either  extremity  of  it  a  line,  equal  and  parallel  to 
b^.  Join  the  extremities  of  these  lines,  and  call  the  closing 
side  c'y\  Consider  them  all  as  positive  as  we  pass  around 
the  triangle  in  the  direction  of  +  aa  and  +  b/S,  and  we  have, 

(330), 

aa  +  b/3  +  c'y'  =  0, 

subtracting  from  the  given  equation  gives 

cy  —  c'y'  =  0, 


248 


QUATERNIONS. 


[339,  340. 


or 


r  =  -r; 


hence,  (327),  the  closing  side  will  be  parallel  to  the  third 
vector,  and  the  relation  cy  =  c'y'  shows  that  they  will  be  of 
the  same  length,  which  was  to  be  proved. 

339.  Three  non-planar  vectors,  cannot  give  the  rela- 
tional' -\-  hft  +  cy  =0,  unless  aa  =  0,  h/i  —  0,  cy  —  0. 

For  a  closed  triangle  cannot  be  made  from  the  three  vec- 
tors ;  hence,  each  term  must  be  separately  equal  to  zero.  It 
does  not,  however,  follow  that  each  vector  is  zero,  for  the  co- 
efficients, a,  h,  c,  may  be  the  algebraic  sum  of  several  num- 
bers ;  a,s,  a  =m  +  n  +p  +  etc.  =  0. 

It  follows  from  the  above  that  if  the  sum  of  three  vectors 
is  zero,  they  must  be  co-planar. 

340.  If  three  co-initial  vectors  give  the  relations 


and 


aa  +  bj3  +  cy  =  0,         (1) 
a  +  &  +  c  =  0,        (2) 


tJiey  will  terminate  in  a  right  line. 

Multiplying  the  second  equation 
by  y  and  subtracting  from  the  first, 
will  give 


a{a-y)  +b{^-y)  =  0, 


Fig.  833. 


or 


a-y=--{/3-y). 


(3) 


But  from  the  figure,  we  have 

a=  y  +  CA, 
and  J3=y+CB; 

.'.  a  —  y=  CA, 
^-y=CB; 
which  in  (3)  gives 

CA  =  -~  CB; 
a 


840.]  CO-FLAXAR    VECTORS.  249 

hence,  (327),  GA  and  C B  are  parallel,  and  since  they  have  a 
point  C  common,  they  coincide  ;  hence  they  terminate  in  the 
same  right  line. 

(Obs. — Since  CA  and  CB  are  co-initial  it  would  at  first  appear  that  they 
ought  to  have  the  same  sign  in  the  result.  But  an  examination  of  equa- 
tion (3)  shows  that  one  or  two  of  the  coefficients  must  be  negative  while  the 
other  two  or  one  is  positive.     If  a  and  h  have  contrary  signs  then  we  have 

b 
CA  =  —  CB.     The  first  equation  shows  that  the  three  vectors,  aa,  h(i,  cy, 

taken  in  succession,  will  make  a  closed  triangle.) 


APPLICATIONS. 

1.  The  extremities  of  two  adjacent  sides  of  a  parallelogram 
and  the  middle  point  of  the 

diagonal  betiueen  those  sides 
are  in  a  right  line. 

Let  AB^a,  AD  = 
ft=  BC,  AE  =  I  AG 
=  6. 

We  have 

AG=AB+  BG, 
or  2(y  =  «:+/?; 

.-.  2d  -a-  ft  =  Q. 

But  the  coefficients  of  S,  a,  ft,  give 

2-1-1  =  0; 

hence,  (340),  the  extremities  B,  D,  and  E  of  the  vectors 
are  in  a  right  line.  This,  then,  is  another  mode  of  prov- 
ing that  the  diagonal  BD  bisects  AG.  Similarly,  AG  bi- 
sects BD. 

2.  The  medials  of  a  triangle  meet  in  one  point. 

This  has  already  been  solved,  (337,  2),  but  we  here  pre- 
sent another  demonstration.     In  Fig.  232,  we  will  show  that 


250  QUATERNIONS.  [340. 

the  extremities  of  the  vectors  AL,  AO,  AC,  are  in  a  right 
line.  If  they  are  in  a  right  line,  we  must  find  values  for  a, 
b,  c,  which  will  satisfy  the  equations 

aAL  +  bAO  +cAa=0,  (1) 

a  +  b+  G  =  0.  (2) 

Substituting  the  values  AL  =  ^y,   AO  =  ^  {y  —  /3),  AQ 
--  §,  gives 

lay^\b{y-0)-cp=  0, 

or  (3a  +  W)y  -  (26  +  6c)yS  =  0.  (3) 

But  /?  and  y  being  independent  we  must  have 

3a  +  2&  =  0, 
26  +  6c=0; 

from  which  we  have 

<^  =  —  I  &,  and  c  =  —  5  6 ;  (4) 

or  they  are  indeterminate.  Substituting  these  values  in  the 
second  equation,  gives 

-%^  +  h-\b, 

which  reduces  identically  to  zero ;  hence  every  value  of  b 
will  reduce  (1)  and  (2)  to  zero,  and  the  points  L,  0,  C,  are  in 
a  right  line.  Making  &  =  —  3  in  (4),  gives  a  =  2,  and  c  =  1, 
and  (1)  and  (2)  become 

2^i:-3^0  +^C=0, 
2-3+1  =  0. 

3.  The  dltitvdes  of  a  triangle  rneet  in  a  common  point 

(The  altitudes  are  the  perpendiculars  from  the  angles  upon  the  opposite  * 
sides.) 


340.] 


CO-PLAXAR    VECTORS. 


251 


Let  AE,  BF,  CP,  be  the  respective  perpendiculars,  the 
lengths  of  the  sides  he  I—  AB,  q 

m  =  BC,  71=  CA,  ;'i,  a,,  A, 
unit  vectors  along  the  corre- 
sponding sides.  Let  the  direc- 
tions be  positive  as  indicated 
in  Fig.  232  ;  then  we  have 

a  =  ma^ ,     ft  =  nfti ,     y  =  ly^. 

J^O  =  rjui.  As  lines  we 
have 


AP  =  n  cos^, 
AF  =  I  cos  A  ; 


CP  =  71  sin.  A, 
BF  =  I  sin  A, 

and  as  vectors 

PC  =71  sin  A.  e^, 
BF  =  I  sin  ^.  ywi, 

We  have  from  A  APC, 

AP  +  PC  +  CA  =  0, 

or  n  cos  A.  y^  4-  w  sin  A.  e^  +  Tiftx  =  0 ; 


AP  =  n  cos  A.  yi, 
FA  =  1  cos  A.  ft^. 


_        /?i  +  cos  A  .  yx 


f  1  =  — 


sin  A 


and  from  A  ABF 


vi  +  cos  A .  /?i 
/^i  =  - 


sin  J.         * 
From  ABOCA,  we  have 

ln  +  rMi  +  qh  +  nftx=0; 

in  which  substituting  the  values  of  fi  and  Mi,  collecting  terms. 


252  QUATEEmONS.  [840. 

and  making  the  coefficients  of  y^^  and  y?i  separately  equal  to 
zero,  we  find 

I—  n  cos  A 

*"  =  • A » 

sm  J. 

_n  —  I  cos  A 
^~       sin  A       * 

The  triangle  ABO  gives 

_  ,         (n  cos  A  —l)  (yi  +  cos  A .  /3i) 

=  ^-j  [(^  cos  A  -  1)13^  +  {n-lco^A)y,  ].   (1) 

We  now  find  that  P,  0,  C,  are  in  a  right  line,  for  if  in 
the  equation 

aAP  +  hAO  +cAC=0,  (2) 

we  substitute  the  value  of  ^  0  from  equation  (1),  and  re- 
duce, we  find 

_  cos  A  {n  cos  A —  I), 

C -.    s     'i  Of 

n  BUT  A 

I  cos  A  —  n, 

a  = .  ^   A     o; 

n  sm-  A 

.'.  a  -:-  6  -f-  c  =  0; 

which  was  to  be  proved;  therefore  the  altitudes  of  a  triangle 
meet  in  a  common  point. 

(It  will  be  found  best  to  get  the  value  of  vector  ^0  in  terms  of  the  ad- 
jacent vectors,  y  and  /?,  and  the  included  angle  A,  as  has  been  done  in  the 
preceding  solution.) 

EXAMPLES. 

1.  Three  co-initial  vectors  2a,  3/3,  cy,  terminate  in  a  right  line  ; 
find  c. 


840.] 


CO-PLANAR    VECTORS. 


253 


2.  Three  co-initial  vectors  7a,  —  4/5,  3a,  are  connected  by  tlie  rela- 
tion 2/S  =  3/5  —  a ;  do  they  terminate  in  a  right  line  ? 

3.  The  perpendiculars  to  the  sides  of  a  triangle  from  the  middle 
points  meet  in  one  jDoint.  (The  point  will  be  the  centre  of  the  circum- 
scribed circle.) 

Let  FO"  be  perpendicular  to  ABat  its  middle  point,  EO"  perpendic- 
ular to  ^C  at  its  middle  point;  then  if 
a  line  from  0"  to  the  middle  point  of 
BG  is  perpendicular  to  that  side,  the 
problem  is  true.  We  have  to  prove 
that  vector  0"D  is  a  multiple  of  the 
vector  from  A  periDcndicular  to  CB, 
the  value  of  which  is  given  by  equa- 
tion (1)  in  the  third  problem  above. 

4.  To  find  vector  A  O'"  drawn  from 
the  angle  A  to  the  centre  of  the  cir- 
cumscribed circle. 

Solution. — Letting  the  signs  of  the  Fig.  236. 

vectors  be  represented  as  in  Fig.  232,  we  have  from  the  preceding  figure, 

Vector  OG  —  —  n  cos  A  .y^—  nfiy , 

Vector  BE '  =  —  I  cos  A  .  /Ji  —  ly^, 
(where  BE'  is  an  altitude  from  B).     The  polygon  AFO"'EA  gives 

AF  +  FO"'  +  0"'E  +  EA-Q^ 
or         \lyi  —  xn  (JJi  +  cos  A  .  y  ^)  —  yl  {y^  H- cos  A  .  /3,)  +  infS^  =  0. 

Exjjanding,  collecting  terms,  and  equating  to  zero,  separately  the 
coefficients  of  /?,  and  ;^,,  we  find 

n—l  cos  A 


2n  sin**  A   ' 

(71  —  I  cos  A)  (n/?,  +  n  cos  A  .  y^) 


Then 


F0'"=  - 


AG'  =AF+F0" 


2n  siu^  A 


(n  —  l  cos  A)  {nfi^  +  n  cos  A.y^) 
=  ^^y^-  2n  sin*  A 

=  2  sin«  A  ^^  ^°^  A-n)  fi^+Q,-n  cos  ^)ri], 


which  is  the  quantity  sought. 


254 


qUATEBmONS. 


[341,  342. 


5.  If  tlie  sides  of  two  polygons  are  parallel  and  lie  in  the  same  di- 
rection, their  sides  are  proportional. 

6.  If  the  corresponding  vertices  of  two  triangles  are  in  lines  radiat- 
ing from  a  point,  the  corresponding  sides  prolonged  will  meet  in  a  right 
line.     (This  is  solved  by  Transversals,  further  on.) 


Angle-Bisectors. 

341.  A  line  which  bisects  an  angle  of  a  triangle  is  called 
an  angle-bisector. 


342.  Problem. 


To  find  the  vector  of  an  angh-hisector. 

Let  CB  be  a,  CA,  (i,  and 
CD,  the  vector  which  bisects 
their  angle,  be  d.  Take  CF 
=  o-i ,  a  unit  vector,  GE  = 
fix,  and  complete  the  rhom- 
bus GEGF,  then  will  EG  = 
ai ,  and  FG  =  pi ;  and  all  be- 
ing positive /rom  G,  we  have 

GG  =  GF+  FG, 

or         GG=  ai  +  pi. 


But  the  bisecting  line  GD  may  be  of  any  length ;  hence 
we  have 

GB  =  xGG, 
or  S  =  x{a,  +  p,),  (1) 

in  which  x  is  indeterminate.  This  is  a  general  expression 
for  the  required  vector.  If,  however,  the  vector  CD  is  lim- 
ited by  the  line  which  joins  the  extremities  of  two  given 
vectors,  its  value  becomes  determinate.     For  we  have 


or 


GA  +AB-GB  =  0, 
nfii  +  lyi  —  max  =  0, 
mai  —  n/3i 


n  = 


I 


342.]  ANGLE-BISECTORS.  255 

Also  6  =  x{a^^-  /3,)  =  CA  +  AD 

=  n^i  +  yn 


Substituting  y^, 

=  nfti  +  ^  (mo'i 

-nA) 

reducing, 
transposing, 

_n{l-y)ft,  + 

mya^^ 

I 

{Ix  -  my)  a^ ) 

[  =  0; 

} 

+ 

{Ix- 

-n(l-y))p,) 

.-.  (334), 

Ix  —  my  =  0, 

Ix  —  n{l  —  y)  =  0; 

from  whicli  we  find 

mn 

(2) 

m  -^  n 

rd 

V  = • 

m  +  n 

(3) 

Substituting  the  value  of  x  in  (1)  gives 

m  +  n^         '^'^' 

(4) 

which  is  the  definite  value  of  the  vector  which  bisects  the 
angle  between  /nai  and  n/3i ,  and  limited  by  the  line  joining 
their  extremities. 


APPLICATIONS. 

1.  The  medial  to  the  base  of  on  isosceles  triangle  is  an  angle- 
hisector. 

Since  the  triangle  is  isosceles,  m  =  n.  The  medial  to  the 
base  will  be,  (336), 

6=  ^  (mai  +  w/?j) 

(since  m  =  n),  =^m(ai  +  A).  (5) 


256  QUATERNIONS.  [842. 

The  angle-bisector  is  given  by  equation  (4)  by  making 
m  —n,  and  hence  is 

(J  =  1  TO  (o'l  +  A) ; 

which  being  the  same  as  (5)  shows  that  the  medial  bisects 
the  vertical  angle. 

2.  The  angle  loliich  bisects  the  vertical  angle  of  an  isoscdes 
triangle,  bisects  the  base. 

When  CD  bisects  the  angle  C,  the  value  of  AD  ia  given 
by  equation  (3),  which  gives,  when  m  —  n, 

that  is,  AD  —  ;^  AB,  which  was  to  be  proved. 

3.  Any  angle-bisector  of  a  triangle  divides  the  opposite  side 
into  parts  proportional  to  the  adjacent  sides. 

When  CD  bisects  C,  we  have  found,  (Eq.  (3) ), 

AD  =y 


.'.  DB=l^y  = 


m  +n 
rrd 


m  +  n 
Dividing,  we  have 

AD  _n_CA 
DB  ~m  ~  CB' 

which  was  to  be  proved. 

4   To  find  the  conditions  which  will  cause  the  diagonxd  of  a 
paraJldogram  to  bisect  the  angles  through  which  it  passes. 

We    have    oc  = 


D 

0 

ma\ ,  (3  =  m^i ;  and 

fi/ 

^\7 

^-^ 

6  =  a  +  /3 

6^^^><E^ 

^4 

=  mo'i  +  n^i ; 
and  if  S  bisects  the 

a 

angle,  we  have 

Fig.  238. 

d  =  x{ai+  /i,)  ; 

C42.] 


ANOLE-BISECT0B8. 


257 


therefore  we  must  have 

xay  +  a;/?i  =  wo'i  +  w/?i , 
or  {x  —  m)a'i  +  (,/'  —  n)  /?i  =  0  ; 

.*.     (334),  a;  =  m,  and  x=  n, 
or  w?-  =  w, 

that  is,  the  adjacent  sides  must  be  equal,  and  hence  the  figure 
must  be  a  rhombus. 

5.   The  angle-bisectors  of  a  triangle  meet  in  a  point. 

Let  A3I,  BN,  CL,  bisect  the  respective  angles  A,  B,  C; 

then   considering   the   vectors 

....  ^ 

positive  in  the  directions  indi- 
cated in  Fig.  232,  we  have 

AM=x{y,-  /3,), 
BN^y{a^  -  ri), 

^if  and  5iV intersect  in  0  ; 
draw    CO;    then  if   CO   is   a 
multiple  of  CL,  the  latter  will 
pass  through  0.     AO  will  be  a  multiple  of  AM,  and  ^0  of 
BNy  hence 

^0  =  M(ri- A), 

BO  =  ■?;  (^1  —  >^i). 

We  have  from  the  triangle  ABC, 

lyi  +  ma^  +  n^i  =  0  ; 


n  =  - 


(1) 


Also 


or 


A0-\-  OC  +  CA  =  0, 

.-.  CO  =  n^i+u  (ri  -  A) 

—  mai  —  nftx 


=  nA  +  «(- 


Z 


,) 


17 


=  y  [  (Zw  —  WW  —  ?m)  a  —  mua^.    (2) 


258 

QUATERNIONS. 

Also 

OC  +  CB  +  BO  =0, 

or 

OC- 

-  ma^  +  V  (o'l  — xi)  =  0  ; 
CO  =  —  mo'i  +  v{ai  —  yi) 

,  ,   /     ,  moil  -{-n/SA 

[842. 


=  j[{lv  +  mv  —  ml)  a-i  +  nvySJ.     (3> 

Making  these  values  of  CO  equal,  and  resolving  accord- 
ing to  Article  334,  we  have 


from  which  we  find 


Iv 

+  mv  - 

-  ml  +  mu  =  0, 

.1 

nv  —In 

+nu  +  Zm=  0; 

u. 

u  = 

I 

n  = 

I  +m-\-  n 

v  = 

I 

m^ . 

I  +  m  +  n' 
Substituting  the  value  of  w  in  equation  (2)  gives 

mn  ^y 


z{l  +  m  +  vi) 


hence,  (327),  CO  and  CL  are  parallel  and,  having  the  point 
C  common,  they  coincide. 

The  point  0  is  the  centre  of  the  inscribed  circle. 


EXAMPLES. 

1.  Prove  that  the  line  which  bisects  the  external  angle  of  a  triangle 
divides  the  opposite  side  (prolonged)  into  segments  proportional  to  the 
sides  about  the  angle  bisected. 

2.  In  Fig.  239,  if  Mm  be  drawn  parallel  to  BN^  show  that  it  bisects 
CO. 

3.  If  a  line  be  drawn  from  L  parallel  to  BO  it  will  bisect  A  0.  Let 
n  be  the  point  of  bisection,  show  that  mn  will  be  parallel  to  A  G. 

4-  The  centre  of  the  circumscribed  circle,  the  common  point  of  the 


343.] 


TRANSVERSALS. 


259 


altitudes,  and  the  common  point  of  the  medials  of  a  triangle,  are  in  a  right 
line. 

Solution. — We  have  the  following  values  for  the  vectors  from  A  to 
the  several  points  : 

Fig.  236,        AO"'^   2  sin^  ^  [  (^cos^-  n)  (3,  +  {I-  n  cos  A)r,-\ 

cos  A 
Fig.  235,         AO  =  -^-rj_\.{n  cos  A  -  l)  fi^  +  (n  -  I  cos  A)ri] 

Fig.  239,        AO'  =ii{lr,-nfS,), 

Substituting  y  for  {I  cos  A  —  n),  and  s  for  {I  —  n  cos  A) ;  then,  accord- 
ing to  Article  236,  we  must  have 

a  ,      h  cos  A  c 

which  reduced  gives  a  =  26,  and  c  =  —  36 ; 

.-.  2A0'"  +  AO  -3^0"=0. 
2+1-3=0. 
Hence  the  three  points  0,  0",  0'" ,  lie  in  one  straight  line. 

Transversals. 

343.  A  Transversal  is  a  right  line  which  cuts  a  system 
of  lines. 

The  transversal  of  a  triangle  is  a  line  which  cuts  the  three 
sides,  the  sides  being  considered  as  prolonged  indefinitely. 
The  transversal  may  cut  all  the  sides  on  the  prolonged  parts. 

A  segment  of  a  side  is  the  distance  from  the  point  where 
it  is  cut  by  the  transversal  to  an  angle  of  the  triangle,  meas- 
ured along  the  side. 

APPLICATIONS. 

1.  A  transversal  divides  the  sides  into  segments  such  that  the 
prodiict  of  three  non-  c 

conterminous  segments 
equals  tlie  produx^t  of 
the  other  three  segments. 

Let  BN  =  IAN, 
BM  =  mMC,  CL  = 
nLA,  MC  =  (y,  LA  = 
p,  AN  =  y. 

We  have  from  the 
triangle  ^5(7,  '^  ^^^^^ 

{l-l)y  +(l  +  m)a  +  {l  +  n)/3  =  0;  (1) 


260                                        QUATERNIONS.  [344,  S45. 

triangle  MCL,                  a+n/3—ML  =  0;  (2) 

triangle  ANL,                   NL  +  /3  +  yz=0;  (3) 

also                                                   NL  =  x3IL.  (4) 

Eliminating  between  equations  (1),  (2),  (3),  and  (4)  gives, 
(334), 

BN      BM  CL 


"  AN  ~  MC  LA  ' 
or  BN .  MC  .  LA  =  AN .  BM .  CL, 

which  was  to  be  proved. 

344.  Conversely. — If  three  points  he  taken  on  the  sides  of 
a  triangle,  one  being  taken  on  a  side  prolonged,  or  on  all  three 
prolonged,  dividing  the  sides  into  parts  such  that  the  product 
of  three  non-conterminous  parts  equals  the  product  of  the  other 
three,  the  points  will  he  in  a  right  line. 

345.  The  three  angle-transversals  of  a  triangle  through  a 
oommxm  point,  divide  the  sides  so  that  the  product  of  three  nmi' 
conterminous  segments  equals  the  product  of  the  other  three  seg- 
merds. 

(The  point  may  be  within  or  without  the 
triangle.) 

Let 

AO  =  xd,  OC^ye,BO-=zM, 
and 

LB  =  lAB,  MC  =  mBC,  AN=  nAO. 
-^  From  the  triangle  AOC  we  will 

Bget 
FiG.241.  xd  +  ys  +  ^=0; 

but  since  d  =  y  +  {1  —  m)a,  e  =  a  +  ly,  ^  —  —  a  —  y, 
we  have  x{y  +  {1  —  m)  a)  +  y  (^a  -\-  ly)  —  a  —  y  =  0, 
or  xy  +  x{l  —  m)a  +  ya  +  lyy  —  a  —  y  =  Q', 

.-.  a?  (1  —  m)  +  2/  —  1  =  0  =  a?  +  ?^  —  1, 
which  gives  mx  =  (l  —  Tjy.  (1) 

Similarly  from  the  triangle  BOG  we  will  obtain 

ly  =  {l-n)z,  (2) 


346-349.]    SOLUTIONS  BY  MEANS  OF  TRANSVERSALS.        261 
and  from  BOA  ws  =  (1  —  m)  x.  (3) 

Multiplying  these  three  equatiops  together  and  dividing 
through  by  xy:^,  gives 

Imn  =  (1  -  Z)  (1  -  m)  (1  -  n). 

Substituting  the  proper  values  and  reducing  we  finally 
have 

LB  .  3IC  .  NA=  AL.BM.  CN, 

which  was  to  be  proved. 

346.  Conversely. — If  a  point  he  taken  on  each  of  the  sides 
of  a  triangle,  or  on  one  side  and  two  prolonged,  such  that  the  pro- 
dtLct  of  three  non-conterminous  segments  equals  the  product  of  the 
other  three,  tJien  ivill  the  angle-transversals  to  those  points  pass 
through  a  common  point. 

347.  The  coordinates  used  in  quaternions  are  the  lines 
of  the  figure,  and  such  auxiliary  lines  as  are  necessary  in 
order  to  solve  the  problem.  There  is  not  a  set  of  fixed  lines 
used  as  axes,  as  in  the  Cartesian  system. 

348.  Remark. — Several  of  the  solutions  given  in  this  chapter,  are  much 
longer  than  by  other  well-known  methods  ;  and  the  reader  may  be  inclined 
to  say  that  if  this  feature  is  characteristic  of  the  system,  it  must  be  chiefly 
valuable  for  its  novelty  rather  than  for  its  use.  But  the  merits  of  any  sys- 
tem of  analysis  should  not  be  judged  from  its  ability  to  solve  elementary 
problems,  whether  the  solutions  be  long  or  short.  The  most  powerful  analy- 
sis generally  appears  to  a  disadvantage  when  applied  to  such  problems. 
Were  the  processes  of  the  Calculus  exhibited  to  the  student  in  solving  such 
problems  only  as  :  The  shortest  distance  between  two  points  is  a  straight 
line  ;  The  evolute  of  a  circle  is  a  point ;  The  shortest  distance  between  two 
points  on  the  surface  of  a  sphere  is  the  arc  of  a  great  circle  ;  the  student 
might  infer  that  it  was  a  cumbersome  and  tedious  process  of  proving  what 
could  very  easily  be  learned  by  very  simple  means,  instead  of  the  most  pow- 
erful system  of  analysis  ever  devised.  One  merit,  at  least,  is  apparent  in 
the  preceding  solutions  ;  it  furnishes  an  uniform  method  for  the  solution  of 
this  class  of  problems. 

Solutions  hy  Means  of  Transversals. 

349.  By  discussing  a  general  property,  all  the  particulars  contained  in 
it  may  be  deduced.  We  here  present  a  solution  of  several  of  the  problems 
already  given,  by  deducing  them  from  the  proposition  contained  in  Arti- 
cle 343. 

1.  To  prone  the  proposition  contained  in  Article  345. 


262 


qUATEBNIONS. 


Fig.  242. 


Let  0  be  the  point  througli  which 
the  angle-transversals  are  drawn.  Let 
BF  be  a  transversal  to  the  triangle 
ACP,  then  we  have,  (343), 

PB^  OU  .  FA  =  AB  .  PO  .  CF; 

and    triangle    CPB,   transversal   AE, 
gives 

AB  .  EC .  OP  =  AP  .  BF .  OC. 

Multiplying  together  and  cancel- 
ling common  factors  gives 

PB.  EG  .FA  =  AP  .  BE.  CF, 


which  was  to  be  proved. 

2.  The  medials  of  a  triangle  meet  in  a  point. 

For  we  have 
C 

AL  =  LB. 

BM=  MO, 

CN=NA; 

and  multiplying  together  gives 

AL  .  BM.  CN=LB,  MC .  NA, 
J 

which,  according  to  Article  846,  shows  that 
the  lines  AM,  BN,  CL,  meet  in  a  point. 

3.  T?ie  angle-bisectors  of  a  triangle  meet  in  a  point. 
We  have 

CBLB       AC  _  MC        AB      NA 
CA-  AL'     AB-  BM  '     BC  "  CN  ' 
and,  multiplying  together,  we  get 

AL  .  BM.  CN=LB  .  MC.  NA  ; 

which,  according  to  Article  346,  is  the 
C  required  result. 

4.   The  altitudes  of  a  triangle  meet  in 
a  point. 

From    the  similar    right    triangles 
CPB,  AEB,  we  have 

CP  :  PB  :  :  AE  :  BE. 
Similarly,  BF  :  FA  :  :  CP  :  PA, 
also  AE  :  EC  :  :  BF  :  FC; 

multiplying  and  dropping  common  fac- 
tors, we  find 

PB  .  EC .  FA-  BE .  CF.  AP; 


Fia.  244. 


350.]         SOLUTIONS  BY  MEANS  OF  TRANSVERSALS. 


263 


hence,  (346),  they  have  a  common 
point  0. 

5.  If  the  corresponding  ver- 
tices of  two  triangles  are  in  lines 
radiating  from  a  point,  the  cor- 
responding sides  prolonged  will 
meet  in  a  right  line. 

Let  ABC,  ABC"  be  two 
triangles  having  their  vertices  in 
lines  radiating  from  0,  then  will 
the  intersections  of  AB,  A'B'  ; 
EG,  B'C  ;  AC,  A  'C,  be  in  the 
line  NL. 

We  have  from 

triangle  OCA,  secant  C A' . . 
OCB,  "  C'B'.. 
OB  A,      "      BA'.. 


Fig.  245. 

.  OC  .  CM.AA'  =  CC\AM.OA\ 
.CC.BL.OB'^OC'.CL  . BB'. 
.BB'  .AN.  OA'  =  OB'  .BN.AA'. 


Multiplying  together  and  cancelling  common  terms,  gives 

CM.  BL  .  AN^AM.  CL  .  BN ; 

hence,  (344),  N,  M,  L,  are  in  a  right  line. 

350.  A  complete  quadrilateral  is  a  four-  E 

sided  figure  in  which  each  side  cuts  all  the 
others,  the  sides  being  prolonged  indefinite- 
ly. Thus,  A  BCD  is  a  complete  quadrilateral, 
the  side  BC  cutting  the  other  sides  in  B,  C, 
E;  AB  in  the  points  B,  A,  F;  AD  in  A,D, 
E ;  and  CD  in  C,  D,  F.  A  line  which  is  not  a 
side,  joining  any  two  angles  of  a  quadrilateral, 
is  a  diagonal.  There  are  three  diagonals,  AC, 
BD,  EF.  There  are  also  three  quadrilate- 
rals in  the  complete  figure  ;  the  common  con- 
vex ABCD,  the  uni-concave  EDFB  ;  and  the 
bi-concave  EDFADC,  composed  of  two  oppo- 
site triangles. 


EXAMPLE. 


Tlie  middle  points  L,  M,  N,  of  the  diagonals  of  a  complete  quadri- 
lateral are  in  a  right  line. 


CHAPTEE  n. 

MULTIPLICATION   AND    DIVISION    OP    VECTOBS. 

[Remabk. — The  first  time  that  a  student  reads  the  following  chapter, 
he  will  naturally  be  led  to  ask.  What  right  has  one  to  make  such  assump. 
tions,  What  led  Hamilton  to  think  of  it,  and,  Of  what  use  can  it  be  ?  We 
cannot  answer  the  first  question  more  concisely,  and,  at  the  same  time, 
comprehensively,  than  in  Hamilton's  words.  He  says  :  "  If  the  knowledge 
previously  acquired,  by  any  suitably  performed  analysis,  be  afterwards 
suitably  applied,  by  the  Synthesis  answering  to  that  Analysis,  it  will  conduct 
to  a  suitable  result.  Or  that  whatever  has  been  found  by  Analysis  may 
afterwards  be  used  _by  Synthesis  ;  and  that  the  thing  which  is  repro- 
duced by  this  synthetic  process,  will  be  the  same  with  that  which  had  been 
submitted  to  analysis  previously." — (Hamilton's  Lectures  on  Quaternions, 
p.  38.) 

As  a  partial  answer  to  the  second,  we  will  give,  in  an  Appendix,  a  brief 
sketch  of  the  history  of  the  invention.  It  is  unnecessary  to  answer  the 
third,  for  the  reader  will  soon  see  how  the  method  can  be  used  in  the  solu- 
tion of  problems.  The  problems  here  solved  are  simple,  and  are  intended 
only  to  familiarize  the  reader  with  the  use  of  the  symbols,  and  not  as  a  test 
of  the  merits  of  the  system.] 

Division  *of  Vectors. 

351.  Notation. — Conceive  three  rectangular  unit  vec- 
tors, i,j,  k,  radiating  from  a  common 
point  0.    Let  i  be  positive  to  the  right 
y^T^         of  0,  j  positive  upward,  and  Jc  posi- 
tive in  front  of  the  plane  of  ij,  and  the 
"i,     opposite   directions  negative.      Let  a 
left-handed  rotation  be  positive  ;  thus 
if  one  is  looking  along  the  positive 
vector  k  toward  the  origin,  then  will 
^**  ^^'  the  rotation  of  i  upward  toward  j  be 

264 


O 


852-354.]  BECTANOULAB  VECT0B8.  265 

positive,  and  in  the  opposite  direction,  negative.  Similarly, 
the  positive  rotation  of  j  will  be  toward  the  front  of  the 
plane,  and  that  of  k  will  be  downward. 

352.  Operation. — Conceive  that  J  is  attached  to  i  so  that 
as  i  is  turned,./  may  be  turned  in  either  direction  about  i  as 
an  axis.  Let  j  be  turned  through  a  quadrant  in  a  positive 
direction ;  it  will  fall  upon  k,  and  the  line  which  originally 
coincided  with  vector  J  will  now  coincide  with  vector  k,  and 
is  thus  said  to  produce  k.  The  whole  operation  is  thus  ex- 
pressed :  i  operating  on  j  produces  k,  where  vector  i  is 
spoken  of  as  tlie  operator.     This  statement  is  written 

and  is  read,  "  vector  i  equals  vector/  divided  into  vector  k" 
or,  more  simply,  "^  equals  j  divided  into  ^,"  or,  "t  equals  k 
divided  by, 7."     This  is  a  vector  equation. 

In  the  same  manner  i  operating  upon  k  produces  —  j  ;  and 
upon  —  j  produces  —  k,  and  so  on,  as  shown  by  the  figure. 

If  i  operates  twice  successively  upon,/,  thereby  turning  it 
positively  through  two  quadrants,  it  will  bring  it  into  the 
position  of  — ,/.  Successive  operations  of  a  unit  vector  upon 
a  perpendicular  one  will  be  indicated  by  a  repetition  of  the 
vector  in  the  form  of  multiplication,  or  by  means  of  an  expo- 
nent. Thus  the  two  successive  quadrantal  rotations  above 
described  will  be  written  : 


■3 


Li  z=  ii  =  {''^  =  — ^  =  -  1,  (2) 

J 

and,  three  successive  quadrantal  rotations  will  be  written : 

%^   =    T-. 

J 

Generally,  if  the  rotation  be  through  t  successive  quad- 
rantal rotations  producing  some  vector  /?,  whose  position 
can  be  definitely  designated  only  when  the  numerical  value 
of  t  is  known,  we  have,  according  to  the  above  plan : 

*'  =  4-  (3)' 


266 


QUATERmONS. 


[853. 


The  analysis  indicated  by  Equation  (3)  may  be  so  extended  as  to  make  it  a 
General  Equation.  Thus,  while  the  base  i  is  a  unit  vector,  it  may  be 
positive  or  negative  ;  for  if  the  rotation  of  j  about  +  i  be  positive,  we  have 
only  to  rotate  negatively  about  —  i  to  produce  the  same  result.  Again,  the 
exponent  t  may  be  uot  only  entire,  and  positive  or  negative,  but  also  frac- 
tional. ' 

Thus,  if  vectors  OB  and  OA  are  equal  in  length  and  perpendicular  to 
vector  i,  the  arc  BA  be  represented  by  t  where  t  is  a  fractional  part  of  a  quad- 
rant (say  one-third  of  a  quadrant,  or  <  =  ^,), 
then  if  vector  OB  be  revolved  in  a  positive  di- 
rection about  +  i  to  coincide  with  vector  OA, 
we  have,  according  to  the  notation  of  Equation 
(3). 


i' 


—  iM  — 


OA 
OB 


Fio.  S48  a. 


If  —  i  be  the  axis  of  rotation,  then  will 
the  direction  of  rotation  of  OB  into  OA  be 
negative,  that  is,  t  will  be  negative,  and  we 
have 

OA 


(-*)-«  = 


OB 


i' 


(-»•)- 


If  OA  be  revolved  about  +  i  to  coincide  with  OB,  t  will  be  negative;  and 
if  about  —  i,  t  will  be  positive,  hence 


OB       , 


iy 


The  two  preceding  equations  show  that  the  value  of  an  expression  is  not 
changed  by  changing  the  signs  of  both  the  base  and  the  arc  t  in  the  expo- 
nent. Employing  in  this  science  the  algebraic  notation  for  the  reciprocal, 
we  have 

._,         1 


i' 


=  (-»)'. 


(4) 


In  the  expression  —  V.,  the  minus  sign  applies  to  the  entire  expression,  so 
that  we  have  the  identical  equation 

-i'  =  -  (i'). 

In  this  case  the  effect  of  the  negative  sign  is  to  reverse  the  result  pro- 
duced by  i'.  Thus,  if  i'  —  OA  -i-  OB  as  given  above,  then  will  —  j'  = 
—  {OA  -=-  OB)  be  equivalent  to  revolving  OB  into  the  position  of  OA,  and 
then  reversing  the  result,  producing  —  OA.  But  this  is  the  same  as  re- 
volving OB  at  once  into  the  position  of  —  OA,  or  of  —  OB  into  OA,  the 
angle  of  which  is  2  +  / ;  or  if  the  rotation  be  about  —  i,  the  positive  angle 
will  be  2  —  <  ;  hence 


~i'=i-  (iO  = 


OA_^OA  OA 

0B~     OB    "  -OB 
If  OA  were  the  divisor,  we  would  have 

*      ~      0A~       f~^ 


p+t  _  y_ 


=  (-if 


(5) 


353,  354.]  THE   VERSOR.  267 

If  the  angle  AOB  be  0  degrees,  then  ^  =  0  -4-  ^  tt  (where  ^  it  represents 
90°)  and  (3)  becomes 

J 

On  the  same  plan,  successive  rotations  equal  respectively  to  x,  y,  s,  etc., 
is  the  same  as  one  rotation  equal  to  the  sum  of  x  +  y  +  z  +■  etc.,  or 

i^ .  '^  .  i' .  etc.  =  4(*  +  y  +  =  +  «»<=■), 

where  the  exponents  follow  the  algebraic  law. 

From  the  above  ifc  appears  that  in  Equation  (3)  the  vec- 
tors i,  j,  ft,  may  be  positive  or  negative,  thaty  and  ft  may  be 
of  any  length,  provided  only  that  their  lengths  are  equal. 

363.  If  the  vectors  of  the  divisor  and  dividend  are  of 
unequal  lengths,  let  ix  and  ft  be  unit  vectors  in  a  plane  per- 
pendicular to  the  axis  i,  a  and  h  their  lengths  respectively, 
and  c  =  b  -^a;  then  if  t  be  the  angle  between  (x  and  ft,  we 
have 

.,      hft      b      ft  ,f,. 

aix      a      ix 

where  the  operation  is  conceived  to  be  that  of  turning  the 
vector  acx  about  the  axis  i  through  an  angle  t,  to  coincide 
in  direction  with  ft ;  then  comparing  the  length  of  6/i  with 
that  of  aa.  The  latter  consists  of  an  algebraic  division  of  h 
by  a. 

If  c  =  d\  we  have 

(di)*  ^  M  .  (7) 

aae 

354.  The  Versor. — A  versor,  literally,  implies  that 
which  turns  about,  and  refers  to  the  agent  producing  the  ro- 
tation. In  this  system,  the  entire  expression  i',  where  i  is  a 
unit  vector,  and  t  an  exponent,  entire  or  fractional,  positive 
or  negative,  is  called  a  versor  ;  and,  when  operating  as  a  quo- 
tient upon  a  perpendicular  line,  its  effect  is  to  turn  that  line  through 
t  quadrants.  If  ^  =  1,  the  rotation  will  be  through  one  quad- 
rant ;  hence,  every  unit  vector,  as  a  versor,  turns  a  perpen- 
dicular line  through  a  quadrant,  and  is  called  a  quadrantal 

versor.  According  to  Equation  (3)  -r  represents  a  versor,  ft 
and  J  being  unit  vectors  perpendicular  to  L 


268  QUATERNIONS.  [355,  356. 

An  operator  is  conceived  to  be  an  agent  which  changes  the  direction  of  a 
line,  by  rotating  it ;  or  its  length,  by  stretching  it.  A  versor  as  an  operator 
rotates  a  line,  and  a  tensor  elongates  it.  In  division,  we  conceive  that  the 
versor  operates  upon  the  divisor  line,  turning  it  into  the  direction  of  the 
dividend  line,  after  which  the  tensor  of  the  quotient  stretches  the  divisor 
line  to  the  length  of  the  dividend  line.   ^ 

Multiplication  of  Vectors. 

355.  Equation  (3)  may  be  written  : 

i'j  =  A  (8) 

where  the  versor  is  the  multiplier,  the  vector  operated  upon 
is  the  multiplicand,  and  is  perpendicular  to  the  axis  of  the 
versor,  and  the  vector  produced  is  the  product,  also  perpen- 
dicular to  the  axis  of  the  versor.  This  mode  of  transforma- 
tion from  division  to  multiplication  corresponds  to  that  of 
clearing  an  algebraic  equation  of  fractions,  but  it  will  soon 
be  shown  that  the  order  of  the  factors  given  above  must  be 
particularly  observed. 

Equation  (8)  is  read  "i^  multiplied  intoj,"  or,  ",/  multiplied 
by  «'  —  (d"  )  but  in  no  case  is  it  read  "  i''  multiplied  by  j." 

If,  in  the  preceding  discussion  of  the  division  of  unit  vectors,  multiplicand 
be  substituted  for  divisor,  and  product  for  dividend,  it  will  be  applicable  to 
this  form  of  multiplication.  If  the  lengths  of  the  vectors  are  unequal,  the 
multiplicand  is  revolved  about  the  axis  of  the  versor  to  coincide  in  direction 
with  that  of  the  product,  and  the  multiplicand  is  then  stretched  the  amount 
required  by  the  multiplier,  thus  producing  the  product.  Thus,  from  Equa- 
tion (6),  we  have 

ci'.  aa  =  bfi. 

This  is  not  an  ordinary  algebraic  *  multiplication  ;  it  is  a  vector  multipli- 
cation. We  are  not,  therefore,  at  liberty  to  apply  it  to  the  rules  of  ordinary 
algebra,  but  must  develop  the  rules  which  govern  the  operations  in  accord- 
ance with  the  principles  upon  which  it  is  founded. 

356.  The  non-cominutative  principle. — Suppose  that 
j  operates  upon  i,  turning  the  latter  left-handed  so  as  to 
cause  it  to  fall  on  —  ^ ;  then  we  have  (Fig.  247), 

ji  =  -k 


*  Ordinary  algebra  treats  of  operations  upon  literal  numbers.  The  ana- 
lytical part  of  quaternions  is  considered  as  an,  algebra.  .  Prof.  Benjamin 
Pierce,  of  Harvard  College,  discovered  163  systems  of  algebra  (see  John- 
son's Encydopiedia,  Article  Qualitative  Algebba). 


357.]  NON-COMMUTATIVE  PRINCIPLE.  269 

If  i  operates  on  j,  we  liave 

ij  =  h, 

which  shows  that  in  the  multiplication  of  rectangular  vec- 
tors, reversing  the  order  of  the  factors  changes  the  sign  of 
the  result.  This  is  one  of  the  most  peculiar  jjrinciples  ot 
this  system.  In  algebra  the  factors  are  interchangeable,  thus 
aft  =  &a,  3  X  4  =  4  X  8,  and  for  this  reason  are  said  to  be 
commutative,  but  in  quaternions  we  have  for  rectangular  vec- 
tors ij  —  —  ji,  and  the  factors  are  non-commutative. 

1l\iq  plus  and  minus  signs  are  also  commutative.  Thus 
i{  —  j)  =  —  ij.  For  if  in  Fig.  247,  —i  operate  upon  j,  it 
will  produce  —  k,  which  is  the  same  as  i  {  —  j).* 

357.  The  associative  principle  consists  in  grouping 
the  factors  in  different  ways,  thus, 

i-jk  =  ij-k  =  ijk.  I 


'J 


If  the  cyclical  order  of  the  factors  be  pre- 
served, the  products  will  be  equal ;  thus  we 
have 

ijk  —  jki  —  kij. 

Fio.  248. 

But  if  the  cyclical  order  be  deranged,  the  sign  of  the 
product  will  be  changed,  thus 

ijk  —  —  ikj  =  —  kji  =  —  jik, 

a  result  which  follows  directly  from  the  preceding  Article. 

EXAMPLES. 
Show,  from  the  figure,  that  the  following  equations  are  correct: 


1. 

ji=  —k, 

j{-k)  =  -i,       j{-i)  =  k. 

jk  =  i. 

2. 

M  =  j. 

Jcj  =  —  i,       k(-i)=-j. 

k{-^j)  =  i. 

8. 

—  ij  =  —  k. 

i{-k)z=j,            -i(-j)=:k. 

—  ik  =j. 

4. 

—  ji=  k, 

—jk  =  —  i,    —j  (—  t)  =  —  k, 

-j{-k)=i. 

5. 

—  M-  —j, 

-  *(-i)  =  -i,  -k(-  i)  -j. 

—kj  =  i. 

6. 

ijk  =  p  =/  =  k^=-l. 

*  There  are,  however,  many  cases  in  algebraic  analysis  where  the  ele- 
ments in  the  expression  are  non-commutative.  Thus  a',  log'  x,  log  sin  x, 
are  not  equal  respectively  to  x»,  log*  3,  sin  log  x. 


270  QUATERNIONS.  [358, 359. 

358.  The  square  of  any  vector  equals  minus  the  square 
of  a  line  of  equal  length. 

Let  a  be  tlie  length  of  the  vector,  then  if  i  be  a  unit  vec- 
tor, the  entire  vector  will  be  ai,  and  we  have  (since  i^  =  —  1, 
(352)), 

{a%){ai)=a''i?=-a\ 

which   was  to  be    proved.      This   furnishes   a   convenient 
method  of  changing  vectors  to  lines. 

359.  The  reciprocal  of  a  unit  vector  equals  minus  the 
same  vector. 

Equation  (2),  (352),  gives 

--M  =  l;  .-.  -i  =  \.^i'K  (9) 

I 

That  is,  &  positive  quadrantal  rotation  about  —  i  as  an  axis,  is  equivalent  to 
Si'negative  quadrantal  rotation  about  -|-  i  ;  but  the  latter  is  called  the  recip- 
rocal of  the  former. 

It  was  observed  (354)  that  /3  -r-^'  represents  a  versor,  and  (355)  that  the 
versor  as  a  factor  is  the  multiplier.  These  principles  lead  to  another  mode 
of  changing  division  into  multiplication  ;  thus,  multiplying  both  members 
of  (S)  into  f  gives 


But,  since  ^"^  =  —  1,  this  becomes 


i'{~l)=  -i'  =  fij  =  -  /3{-j)=  -  ^  .^  (by  m  =  ^  ^  ; 

/3j  =  iK  (10) 


Here  we  have  another  peculiar  result,  that  -while  i'  operating  on  j  as  a 
divisor  produces  /3,  by  turning  the  former  through  an  angle  t,  by  operating 
upon  the  multiplier  —  fi  \t  produces  j,  by  turning  the  former  through  an 
angle  equal  to  2  —  «  ;  that  is,  the  supplement  of  the  angle  t  of  the  versor. 

We  further  observe  that  /J  -i-j  as  a  versor,  operating  as  a  factor  upon  j, 
produces  fi ;  hence,  its  equal ,  —  pj,  must  produce  the  same  result.  We  have 
~  Pj  ■  j>  which,  by  the  associative  principle,  becomes  —  fi .  jj  =  —  fi  .j-  = 
—  /?  ( -  1)  =  /3,  as  it  should. 

Again,  if  /?  be  rotated  about  —  i,  the  positive  angle  will  be  t,  and  its 
supplement  will  be  2  —  t,  hence,  according  to  the  preceding  principle,  we 
write  at  once,  fij  =  (—  t*"*) ,  and  this  compared  with  Equation  (10)  gives 

-t'  =  (-t)«-*. 

The  fact  that  the  versor,  when  operating  upon  the  multiplicand,  is  made 
the  multiplier,  or  is  the  first  one  of  the  two  factors,  determines  the  proper 


860.]  DIVISION  AND  MULTIPLICATION  271 

order  of  the  factors,  so  that  cancellation  may  be  performed  as  if  it  were 
algebraic.     Thus  we  have  (Equation  (3) 

which  is  the  same  as  if  j  were  cancelled  in  the  second  member  by  a  stroke 
inclined  thus  /.  If  written  j-r,  the  result  will  not,  in  general,  be  fi,  and 
the j's  cannot  be  can-^elled. 

360.  From  Equation  (2),  (352),  we  have 

^•2  =  -  1  =  (-  ly  ; 

and,  taking  one-half  the  exponents  of  both  terms,  we  have 

i  =  (-  1)^ 

or,  if  we  employ  the  radical  sign  as  in  ordinary  algebra,  this 
may  be  written 

i  =  V^^, 

and  the  same  result  may  be  found  for  every  vector  which 
operates  to  turn  a  line  through  a  quadrant.  The  expression 
V—  1,  separated  from  i,j,  or  any  other  axis,  may  be  consid- 
ered as  an  iNDETEEMmATE  VECTOK-UNiT,  or  an  unit-vector  with 
indeterminate  direction* 


The  symbol  V  —  1  in  this  system  is,  therefore,  an  analyti- 
cal expression  for  the  turning  of  a  line  through  a  quadrant 
about  an  axis  perpendicular  to  the  line. 

This  is  of  the  same  form  as  the  even  root  of  negative  unity 
in  algebra,  but  the  significations  of  the  two  are  very  differ- 
ent, t  The  V—  1  in  algebra  is  impossible.  It  results  from 
an  attempt  to  find  a  number  whose  square  is  minus  one  ;  but 
the  above  symbol  results  from  expressing  by  an  exponent  a 
foci  which  has  no  representation  in  ordinary  algebra,  com- 
bined with  the  laws  already  assigned  to  the  multiplication 
of  vectors.:}:     The  result  was  reached  by  first  causing  two 

•  Hamilton's  Lectures,  p.  178. 

f  Hamilton's  Lectures,  p.  635. 

X  There  are  several  instances  in  which  tho  same  notation  is  employed, 
having  very  different  meanings.  Thus,  sin-*  x  does  not  mean  the  reciprocal 
of  sin  a; ;  in  the  expression  Fx,  read  "  function  of  a!,"i^i3  not  a  factor. 


272 


QUATERNIONS. 


[361. 


successive  operations  upon/,  producing  i^/  =  —  j,  tlien  going 

backward  one  operation,  producing  i  =  V  —  1  (j  having  been 
dropped).  The  latter  operation  is  a  division  in  this  system, 
and  not  an  algebraic  root. 


Anothar  Method. 

361.  Multiplication  of  Oblique  Vectors. — Let  0A  = 

fi ,  OB  =  a,  each  being  a  unit- vector, 
AOD  =  B  \)Q  the  angle  between 
them  ;  it  is  required  to  find  the  pro- 
duct of  aft  and  /icj.* 

Take  the  unit-vector  e  perpen- 
^  dicular  to  the  plane  of  a  and  (3,  and 
positive  in  front  of  the  plane  ;  and 
vector  j  perpendicular  to  a  and  f. 
The  length  of  OA  being  unity,  we 
have,  as  lines 

DA  =  sin  8, 


+  e 


Fio.  849. 

OB  =  cos  ( 
and  as  vectors,  (328), 

OD  =  cos  B.cy. 


or 


DA  =  sin  d.j. 
The  triangle  ODA  gives,  (330), 

0A=  OD  +  DA, 

ft  =  cos  6.a  +  sin  d.j. 


0) 


Let  a  be  the  multiplier,  then  writing  a  before  each  term, 
we  have 


But  (352), 


aft  =  cos  6,o?  +  sin  d.aj. 


(2) 


*  The  subscripts  to  the  unit-vectors  are  dropped,  and  the  Greek  letters 
are  generally  restored  for  designating  the  vectors.  Hamilton,  for  some 
reason,  used  the  English  letters  i,  j,  k,  to  designate  mutually  perpendicular 
vectors. 


862.]  MULTIPLICATION  AND  DIVISION.  273 

and  aj  =  e; 

.'.  a/3  =  —  cos  6  +  esinO,  (3) 

which  is  the  result  sought. 

If  p  be  the  multiplier,  we  have,  writing  a  after  each  term, 

/3a  =  cos  fttt^  +  sin  O.ja,  (4) 

=  —  cos  6  —  e  sin  <9,  (5) 

which  is  the  same  as  making  £  negative  in  (3). 

362.  Division  of  Oblique  Vectors. — Let  a  be  the  di- 
visor, then  from  (1)  we  have 

/3  i 

-  =  cos  /^  +  sm  6^  ^ , 
a  a 

=  cos  (^  +  f  sin  6.  (6) 

The  same  result  is  found  by  dividing  the  left  member  of 
(5)  by  or^,  and  the  right  member  by  its  equal  -  1. 

Similarly,  dividing  the  left  member  of  (3)  by  /3^,  and  the 
right  by  its  equal  —  1,  gives 

a 

-s  =  cos  ff  —  s  sm  ^,  (7) 

the  right  member  of  which  differs  from  the  right  member  of 
the  preceding  equation  by  the  sign  of  f. 

EXAMPLES. 

1.  Deduce  the  value  of  '—  ,  equation  (7),  by  taking  the  reciprocal  of 

the  second  member  of  Equation  (6). 

2.  Show  that  the  product  of  the  right  members  of  Equations  (3)  and 

(5)  is  unity,  and  therefore,  as  unit-vectors  afi  =  — — ,  (and  not  a/i  =  — — ). 

3.  Find  the  value  of  ^  for  6  =  0°,  45°,  90%  and  180°. 

4.  Show  that  ia/T)(j3y)  =  —  ay. 


274  QUATERNIONS.  [363-364 

363.  Generally,  if  oc  and  yS  are  not  unit-vectors,  we  have 

al3  =  TaTfS  (-  cos  ^  +  £  sin  6) ;  (9) 

fia  =  TftTa  (-  cos  19  -  £  sin  6) ;  (10) 

^=^(cos^-"£sin^);  (11) 

i=^(cose+  £sin^).  (12) 

a      IOC  ^ 

Since ^'  has  disappeared  from  the  result,  or,  more  generally,  since  a  and 
P  are  simply  confined  to  a  plane  perpendicular  to  tlie  axis  e,  it  is  evident 
that  they  may  have  any  position  in  that  plane.  The  axis  e  fixes  the  posi- 
tion of  the  plane  of  the  two  vectors,  but  not  their  position  in  that  plane.  The 
parenthetical  parts  are  versors. 

It  also  appears  that  multiplying  or  dividing  one  vector  by  another  in- 
volves the  three  rectangular  dimensions  of  space. 

364.  Scalar.* — Since  tensor  a  and  tensor  /3  are  each 
numerical,  their  product  or  quotient  will  be  numerical,  and 
when  multiplied  by  cos  6,  the  final  product  will  also  be  nu- 
merical, and  may  be  positive  or  negative.  This  product 
Hamilton  called  the  Scalar  part.  It  represents  the  reals  of 
algebra.  It  is  represented  by  the  letter  S  placed  before  an 
undeveloped  expression  ;  thus,  Sa/3  implies  the  scalar  part 
of  the  product  of  a  into  J3,  and  equals  TaTp  cos  6.  Simi- 
larly for  S— . 


365.  The  Vector  Part.— The  other  part  of  (9),  or 
TaTft  siu  ^.  f,  is  a  vector  either  longer  or  shorter  than  the 
unit- vector  f,  and  similarly  in  regard  to  the  other  three  equa- 
tions.    The  Vector  part  is  indicated  by  the  letter  V  placed 

before  an  undeveloped  expression  ;  thus,  Va§,   V  — ,  etc. 

366.  A  Quaternion  is  the  result  of  multiplying  or  divid- 

*  Latin,  scala,  series  of  steps  ;  so  called  because  it  represents  discontinvr 
0U8  number. 


867.]  MULTIPLICATION  AND  DIVISION.  275 

ing  one  directed  line  by  another  in  space.  This  name  was 
first  applied  to  the  result,  because  the  result,  when  found  by 
means  of  rectangular  vectors,  gave  an  expression  of  four 
t«rms,  of  the  form  lo  +  ix  +  jy  +  Jcz,  where  zv  is  numerical, 
ifj,  k,  mutually  perpendicular  unit- vectors,  and  x,  y,  z,  dis- 
tances along  the  vectors. 

A  Quaternion  is  the  product  of  a  tensor  and  a  versor  (Eqs.  (9)  and  (10)) ;  or 
the  product  of  a  tensor  into  a  unit-vector  with  a  scalar  exponent  (Eq.  (6), 
Art.  353)  ;  or  the  power  of  a  vector  (Eq.  (7),  Art.  353) ;  or  the  power  of  a 
versor  when  the  tensor  is  unity  (Eq.  (10),  Art.  359,  and  Eq.  (7),  Art.  353) ; 
or  the  sum  of  a  scalar  and  vector  (Eqs.  (1)  (2)  (3)  (4)  of  Article  367). 

In  each  of  these  forms  of  the  quaternion,  four  elements  are  involved  : 
the  tensor  (which  may  be  unity),  the  angle  between  the  vectors,  and  two 
angles  for  fixing  the  direction  of  the  unit  axis. 

367.  General  Expressions. — ^We  may  now  write  equa- 
tions (9),  (10),  (11),  (12),  of  Article  363,  as  follows,  disre- 
garding the  algebraic  signs, 

aj3  =  Saft  +  Vafi,  (1) 

^a  =  Sfta  +   Vfta,  (2) 

^  =S-^  +  V-^  ,  (3) 

fi  fi  ft'  ^  ' 

A^sl  +  Vl-;  (4) 

a  oc  a 

and,  by  comparing  these  expressions  with  (9),  (10),  (11),  (12), 
before  referred  to,  we  find 

Saft=-  TaTft  cos  e.  (5) 

Sfta=-  TaT/3  cos  d.  (6) 

5^  =  -^ cos  ft  (7) 

«|=^cosft  (8) 

Vaft^  +TaT ft  smd.s.  (9) 

Vfta  =  -  TaTft  sin  d.s,  (10) 

r^  =  _||ein....  (11) 

r-^=+^^-sme.i.  (12) 

a  la 


:76  QUATERNIONS. 

Comparing  these  expressions,  we  have 


from  (5)  and  (6) 

Sa^  =  spa. 

(13) 

"     (7)  and  (8) 

^ a       (Taf       p 
P~  (Tpf^  a  * 

(14) 

"     (9)  and  (10) 

Vap  =  -  rpa\ 

(15) 

"     (11)  and  (12) 

1 

a  _        {Taf      p 
P            {Tpy      a* 

(16) 

"     (1),  (2),  (13), 

(15), 

ap  +  Pa  =  2Sap, 

(17) 

"     (1),  (2),  (13), 

(15), 

aP-  Pa  =  2  Vap. 

(18) 

368.  Discussion. — 1°.  Let  the  vectors  be  mutually  per- 
pendicular. Then  6  =  90°,  and  (5),  (6),  (7),  (8)  all  reduce  to 
zero ;  hence 

Sij  =  0.  (1) 

Equations  (9),  (10),  (11),  (12),  will  all  reduce  to  the  forms 
given  in  Article  354,  as  they  should. 

2^.  Let  the  vectors  coincide ;  then  ^  =  0  and  the  vector 
part  reduces  to  zero,  and  (1)  becomes 

aP  =  Sap  =  -  TaTP  cos  0  =  -  ah,  (2) 

where  a  and  b  are  the  lengths  of  the  lines. 
3^  Let  S  =  270°,  then 

ap  =  Vap  =  sin  270°.  s  =  -  s,  (3) 

which  agrees  with  the  expression  ii—jj  =  —  k. 

4°.  The  coefficient  of  the  vector  part  of  aP  may  be  rep- 
resented by  a  parallelogram. 

From  equation  (9),  Article  367,  we  have 

Vap=  TaTp  sine.  €. 

Let  Ta  =  a,  Tp  =  h,  then  will  the  coefficient  of  e  be 

l_ 

ah  sin  6.  *  / 

a/ 
Let  a  and  h  be  the  adjacent  sides  of  a  ^- 

parallelogram  ;  then  a  sin  6  will  be  its  alti-  ^*^"  ^^" 

tude  and  ah  sin  0,  its  area. 

5".  The  scalar  of  aP  is  numerically  the  area  of  a  parallel©- 


7 

A 


/ 


)63.]  OBLIQUE  VECTORS.  277 

g  gram  whose  acute  angle  is  the  complement 

7    of  8. 
I  From  (5)  of  Article  367,  we  have 

E  SaP^  ~  T cxT ft  COS,  d 

—  —  (ih  cos  0. 
Draw  A  C  perpendicular  to  ^  I E  and  make 
it  equal  to  h,  and    complete  the  parallelo- 
0  gram  on  a  and  h  as  sides.     Then  will  a  cos  0 

^"^  ^^-  =  a  cos  BAF=  a  sin  .1 BF  =  A F=  the  alti- 

tude of  the  parallelogram  AD;  hence  the  area  of  ABDC 
will  be 

ab  cos  0, 

■which  is  numerically  the  same  as  —  a6  cos  6. 


EXAMPLES. 

1.  Show  that  Sa^  =  —  a«. 

This  is  true  because  a*  =  — «®,  and  the  latter  being  numerical  is  the 
scalar  part. 

2.  Show  that  Fa*  =  0. 

3.  Show  that  Sa  =  0. 

4.  Show  that  Fa  =  a. 

5.  Show  that  {a  +  /3)'  =  cc^  -  2TaT/3  cos  6  +  yS*. 

Solution. — "We  have 

(a  +/?)*  =  (a  +  /?)  (a  +  /?), 
Art.  367,  Eq.  (17),  =  aa  +  25a/?  +  /?*, 

Art.  367,  Eq.  (5),  =a^  -2TaTfi  cos  6  +  /3', 

which  was  to  be  proved. 

6.  Find  what  the  preceding  example  becomes  when  6  =  90°. 

7.  Show  that  (a  -  /5)*  =  a*  -  2Sa/3  +  /?«. 

a 

8.  If  -  =cos  (0—(p)  —  £sin  (Q—tp),  write  the  scalar  and  vector  parts. 

9.  If  77  =  (cos  6  +  £  sin  9)  (cos  <p  +  e'  sin  (p),  find  the  scalar  and  vec- 
tor parts. 

SoLxrrioN. — Expanding  the  second  member,  we  have 

cos  6  cos  <p  +  s  sin  9  cos  q>  +  e'  cos  0  sin  cp  +  ss'  sin  6  sin  <p. 

But,  (361,  3), 

ee'  =  —  cos  (e,c')  +  e"  sin  (e,e')» 


278 


QUATERNIONS. 


in  which  e"  is  a  unit-vector  perpendicular  to  the  plane  of  ee'.     Hence 
we  have 

S~  =  cos  6  cos  qi  —  sin  0  sin  cp  cos  (e.e'), 

/? 
F—  =  £  sin  6  cos  9)  +  £'  cos  9  sin  q)  -f-  e"sin  0  sin  9) sin  (£,«'). 

10.  iS'a/?^  represents  the  volume  of  a  parallelepiped. 
Let  a,  fi,  y,  be  unit-vectors 
along  AB,  A  C,  AE. 

Then,  (Art.  361,  Eq.  (3)), 

afi  =  —  cos  {a,  fS)   +   e  sin 
(a,/3) 

Sa/3y  =  S  [—  cos  {a,/S)  y  +  ey 
sin  (a,  /S)] 

=  Sey  (sin  a,  /5), 

because  S[  —  cos  (a,  /?)];'  =0 
being  a  vector. 
But 

ey  =  —  cos  (s,y)  -f-  e'  sin  (e,x) 

£'  being  a  vector  perpendicular  to  e,  y. 

.'.  Sa/3y  =  8[—  cos  {s,y)  sin  (a,/3)  +  e'  sin  («,;')  sin  (a,/?)] 

=  —  cos  {e,y)  sin  (a,  /5), 

=  —  cos  (90°  —  EAK)  sin  GAB  (EK  being  perpendicular  to 

the  base), 
=  -  sin  ^^JS' sin  GAB. 


Let  the  length  of  the  vectors  be  a,  &,  c,  then 

—  8a/3y  =  abc  sin  GAB  sin  EAK 

=  a .  &  sin  6M5 .  c  sin  EAK 


(1) 


=  ^5.  (72).  EK  {GD  being  perpendicular 

to  ^5), 
=  Volume  ^^C/-J^. 

Since  the  volume  will  be  the  same  whatever  be  the  order  of  the  vec- 
tors, we  have 

Safiy  -  ±  Sayfi  =  ±  Syjia,  etc.  (3) 

If  EAK  =  0,  the  three  vectors  will  be  in  one  plane,  and  we  have 

8afiy  =  0.  (3) 

Conversely,   if  Sa^y  —  0,  neither  a,  yS,  nor  y  being  zero,  the  three 
vectors  will  be  in  one  plane. 


369.] 


OBLiqUE  VECTORS. 


279 


11.  —  1^  Safiy  represents  tlie  volume  of  a  tri- 
angular pyramid. 

12.  Let    a   be   a    vector  perpendicular  to 
y5—  a,  find  the  value  of  Sa(i. 

We  have  from  Article  368,  1% 

Sec  {ft  -  a)  =  0, 
and  expanding,  8{api  _  c^  =  0, 
or  Saft-Sa^^-O, 

or,  (359),  Saft  +  a2  =  0  ; 

.-.  Sa/3  =  -a^, 
which  is  the  result  sought.     The  last  equation  may  also  be  considered  an 
eqvMion  of  condition  of  the  mutual  perpendicularity  of  a  and  (5  —  a. 

369.  We  have  previously  found  (359),  that 

*^=-l,  (1) 

which  shows  that  the  vector  operated  upon  has  been  re- 
versed, and  hence  is  the  measure  of  two  right  angles.  Sim- 
ilarly, 

^-^  =  +  1,  (2) 

indicates  a  revolution  through  four  right  angles,  and  gener- 
ally, if  n  be  an  integer,  we  have 

f4«-2  =  _l,  (3) 

6^'^  =  +  1 ;  (4) 

in  both  of  which  n  is  one  of  the  series  of  natural  numbers, 
1,  2, 3,  etc.,  but  the  exponent  of  the  former  is  an  odd  multiple 
of  2,  and  indicates  n  complete  revolutions  less  two  quadrants, 
and  the  exponent  of  the  latter  is  an  even  multiple  of  2,  and 
indicates  n  complete  revolutions. 

EXAMPLES. 


1.  The  sum  of  the  angles  of  a  triangle 
equals  two  right  angles. 

Solution.— Let  CB=a,  CA—li,  BA—y. 
Let  CA  be  turned  left-handed  about  e  (the 
axis  of  rotation  being  perpendicular  to  CAB) 
to  a  parallelism  with  CB,  or  a  ;  CB  to  a  par- 
allelism with  AB,  ory;  and  AB,  or  —  y,  to 
a  parallelism  with  AC,  or  —  fi.     Or  the  last 


Fio.  254. 


280  QUATERNIONS.  fSTO 

rotation  may  be  right-handed,  coinciding  with  GA  prolonged.  In  either 
case  we  have 

—      a  —       P  —  r 

^     -  ft'       ^     ~r'       ^     -     a' 

Multiplying  together  and  cancelling  common  factors,  we  have 

—  (C+A+B) 

£  =  -1; 

hence  we  have,  (Eq.  (3)), 

or  A  +  B+  C  =  {2n-  l);r-, 

hence,  analytically,  the  sum  of  the  angles  of  a  triangle  may  be  any  odd 
number  of  times  tc\  that  is,  tt,  Stt,  Stt,  etc.  But  this  result  is  obtained 
by  supposing  that  we  pass  around  the  triangle  repeatedly.  Arithmetically^ 
we  pass  around  but  once,  and  the  corresponding  vakie  for  the  sum  of 
the  angles  is  found  by  making  7i  =  1,  and  hence  we  have 

A  +  B  +  (7=:7r=:2  right  angles. 

2.  If  the  sides  of  a  triangle  be  prolonged  in  tlie  same  direction,  the 
sum  of  the  external  angles  will  equal  four  right  angles. 

(Obs. — In  this  case  the  sum  of  the  exponents  of  £  will  equal  4«,  and 
n=l.) 

3.  The  sum  of  all  the  angles  on  one  side  of  a  line  formed  by  lines 
drawn  from  any  point  of  it  equals  two  right  angles. 

4.  If  the  sides  of  a  convex  polygon  be  prolonged  in  the  same  direc- 
tion, the  sum  of  the  external  angles  will  equal  four  right  angles. 

370.  Proposition. — If  an  equation  contains  scalars  and 
also  vectors,  the  sum  of  the  scalars  on  one  side  of  the  sign  of 
equality  iviU  equal  the  sum  of  those  on  the  other  side  ;  and  also 
the  sum  of  the  vector  parts  on  one  side  will  equal  the  sum  of 
those  on  the  other. 

(Obs. — It  should  be  understood  that  the  vectors  are  of  the  first  degree, 
and  that  each  term  contains  only  one  vector,  for  otherwise  the  terms  con- 
taining vectors  may  have  scalar  parts.) 

Thus,  if  we  have 

X  +xa  +  {x  +  z)^  =ax  +  y  ~ b/3, 
then  x  =  ax  +  y^ 

and  xa  Jr  (x  +  %)fi  =  —  6/?. 

For  the  scalars  being  numerical  will  exist  independently 
of  the  parts  which  involve  rotation. 


371,373.]  APPLICATIONS.  281 

EXAMPLE. 
"We  found,  in  Article  369,  that 
cos  (9  +  <p)  +  £  (sin  G  +  <p)  =  (cos  0  +  e  sin  0)  (cos  9)  +  £  sin  9)). 
Expanding  tlie  second  member,  observing  that  f-  =  —  1,  we  have 

cos  (Q  +  q})  +  E  sin  (0  +  (p)  =  cos  5  cos  cp  -f  £  sin  0  cos  cp  +  s  sin  ip  cos  0 

—  sin  &  sin  <p. 

Therefore,  taking  the  scahirs  and  vectors  we  have,  according  to  the 
proposition  (observing  that  £  cancels  out), 

cos  (0  +  <p)  =  cos  G  cos  cp  —  sin  0  sin  <p, 
sin  {0  +  q>)  ■=  sin  G  cos  q)  +  cos  0  sin  cp; 
which  are  well-known  trigonometrical  formulas. 

APPLICATIONS. 

371.  Solutions  of  Plane  Triangles. — We  have  already 
found,  for  a  plane  triangle,  (330),  that  a 

y=a  +  ^.  (1) 

By  operating  upon  this  equation  by       j^/ 
the  multiplication  (or  diA-ision)  of  vectors, 
all  the   cases  of  plane  triangles   may  he       j         ^ 
solved. 

372.  The  sines  of  the  angles  are  proportional  to  the  opposite 
sides. 

In  soMng  this  problem  it  is  necessary  to  introduce  the 
sines  of  two  angles.     Multiplying  both  sides  of  (1)  into  a, 

gives 

ya=  a""  +  /3a,  (2) 

and  taking  the  vectors  of  both  sides,  (see  Ex.  2,  Art.  368, 
and  Eq.  (9),    Art.  367),  we  have 

•      Vya  =  0  +  V^a, 
introducing  tensors,  we  have 

cas  sin  (y,  a)  =  has  sin  (/?,  a), 
dropping  a  and  f ,  c  sin  B  =  h  sin  C ; 

.-.  c  :  6  ::  sinjB  :  sin  0,  (3) 

which  was  to  be  proved. 


282  qUATERmONS.  [373-376. 

373.  To  find  a  side  in  terms  of  the  other  two  sides  and  the 
angles  opposite  those  sides. 

Taking  tlie  scalars  of  equation  (2),  we  have 

Sya  =  —  a-  +  S^a, 
with  tensors,      —  ca  cos  {y,  a)  =  —a'  —ha cos {0,  a), 
cancelling  a  a  =  c  cos  (y,  a)  —h  cos  (/?,  a), 

or  a  =  c  cos  B  +  h  cos  C,  (4) 

the  sign  of  the  last  term  being  changed  because  C,  the  inter- 
nal angle  of  the  triangle,  is  the  supplement  of  the  angle  be- 
tween //  and  a.  The  angles  between  y  and  a,  and  /^  and  a 
are  both  at  the  right  of  the  lines  h  and  c  and  above  a. 

374.  To  find  the  cosine  of  an  angle  in  terms  of  the  sides. 
Squaring  (1),  gives 

y^=a^  +  2Sa^+  /3^,         •  (5) 

or,  (359),  —(^=—a^  —  2ab  cos  {a,  /3)  —  U^ 

or  (?  =  €?  +  2ab  cos  (a,  ft)  +  6^, 

or  (^^a"-  2ah  cos  (7+  h\  (6) 

375.  The  square  of  the  hypothenuse  equals  the  sum  of  the 
A    squares  of  the  sides  about  the  right  angle. 

^  (This  may  be    solved  by  maMng    8a/3  =  0   in    equa- 

tion (5)). 
C 

376.  The  side  adjacent  either  acute  angle  of  a 
right-angled  triangle  equals  the  hypotJienuse  into  the  cosine  of  the 
angle. 

Let  y  =  the  hypothenuse,  then  from  (1) 

/3  =  y  —  a. 

But  /3  and  a  are  mutually  perpendicular,  hence  (see  Ex. 
12,  Art.  368) 

Sa{y-a)=0; 

.'.  Say  =  —a\ 

or,  (367,  Eq.  (5)),  -accosB=  -a^; 

.'.  a  =  ccoaB.  (7) 


377,  378]  APPLICATIONS.  283 

377.   Division  may  be  used  in  the  solution  of  the  same 
examples.     Thus,  dividing  both  members  of  (1)  by  a,  gives 

i'  =  l  +  ^.  (8) 

Taking  the  scalars,  gives 

|eos(::)=l.^eos(f), 

or  -  cos  B    =  1  —  -  cos  C, 

a  a 

which  reduces  at  once,  to  equation  (4). 
Taking  the  vectors  of  (8)  gives 

a  a 

c  h 

or  -  f  sin  5  =  -  £  sin  (7, 

a  a 

which  reduces  to  (3). 

Squaring  (8)  and  reducing,  gives  (5). 


Three  Co-initial  Vectors, 

378.  Throe  vectors  drawn  from  a  point  in  any  direction, 
will,  generally,  be  the  edges  of  a  triedral. 
Let  a^  y^,  y,  be  those  vectors.    We  observe 
that  a  general  relation  between  them  can- 
not be  found  by  addition.     The  process  of  ^^^^       .  , 
triming  a  line  from  a  so  as  to  coincide  in  /?^"--J/  ^ 
direction  with  /S  is,  as  we  have  seen,  ex- 
pressed by  multiplication,  or  by  division. 
Passing  from  a  to  ^,  ft   io  y,   y  io    a,  whether  they  be 
unit-vectors  or  not,  will  be  expressed  by 

ft  y   a^ 

and  as  these,  by  cancellation,  reduce  to  unity,  we  have  the 
equation 

ft  Y  <^     '  ^' 


i^ 


284  •  QUATERNIONS.  [379. 

This  equation  is  as  general  for  triedrals,  as  is  equation 
(1),  Article  330,  for  triangles;  and  by  resolving  this  equation, 
all  the  relations  existing  between  the  facial  and  diedral  an- 
gles of  a  triedral  may  be  determined  ;  and  further,  it  is  appli- 
cable to  the  special  case  in  which  the  three  vectors  are  co- 
planar.  We  will  consider  the  vectors  as  unit-vectors.  The 
following  form  of  the  equation  is  more  convenient  for  use. 

This,  in  the  abridged  notation,  (369),  becomes 

379.  Three  co-initial  co-planar  vectors. — Let  the 
angle  between  a  and  /?  be  ^,  and  between  ft  and  y,  cp ;  then 
will  the  angle  between  a  and  yh^  d  -^  cp. 

The  vectors  being  in  one  plane  and 
unity  in  length,  we  have 

f  =  f  1  =  f 2  > 
and  equation  (3)  gives 

£"  =£"£'', 

Fio.  258. 

which  shows  that  a  rotation  from  a  io  ft  followed  by  a  rota- 
tion from  ft  to  y  equals  a  rotation  from  a.  to  y. 

By  restoring  the  values  of  these  expressions  we  have 

cos(^+<p)  +  f  sin(6'4-'^)  =  (cos  (9  + £sin  ^)(cos^  +  £sin  qj),  (4) 

which  is  the  same  as  the  example  in  Article  370  ;  hence,  ex- 
panding and  reducing  as  in  that  example,  we  have 

cos  (d+fp)  =  cos  6  cos  (p  —  sin  0  sin  <7?, 

sin  (d+<p)  =  sin  0  cos  (p  +  cos  0  sin  ^. 

If  y  were  between  a  and  ft,  we  would  have  6  —  cp  for  the 
angle  between  a  and  y,  and  proceeding  as  before  we  would 
find 

cos  {d  —  cp)  —  cos  6  cos  (p  +  sin  d  sin  q), 

sin  {8  —  (p)  =  sin  6  cos  <p  —cos  ^  sin  ^ ; 


379.]  APPLICATIOA'S.  285 

wliicli  are  the  well-known  trigonometrical  formulas  for  the 
cosine  and  sine  of  the  sum  and  of  the  difference  of  two  arcs. 
Next,  let  0  —  cp,  then  equation  (4)  gives 

cos  2(p  +  £  sm2(p  =  (cos  cp  +  £  sin  fp)-, 
and  continuing  this  operation  n  times,  we  have 

cos  n(p  +  £  sin  ncp  =  (cos  'p  -f  ^  sin  (py\  (5) 

which  follows  the  same  law  as  De  Moivre's  formula.  Sub- 
stituting the  algebraic  expression  for  the  indeterminate  vec- 
tor-unit, (354),  we  have 

cos  nqj  -f-  V—1  sin  nqj  —  (cos  cp  -\-  V  — 1  sin  (j-Y,     (6) 

which  is  De  Moivre's  formula.  It  shows  that  one  rotation 
through  an  angle  wp  is  the  same  as  n  successive  rotations 
through  an  angle  q). 

EXAMPLE. 

Find  the  three  roots  of  unity. 

Putting  equation  (6)  under  tlie  more  general  form 

(cos  q)  +   V —\  sin  q))^  =  cos  n  {2r7C  +  cp)  +  V—1  sin  n  (2r;r  +  (p), 

in  which  r  =  0,  1,  2,  3,  etc.,  and  making  <p  =  0,  and  n  =  ^,  we  have 

/y/l  =  cos  I  rrc  +  ^ —  1  sin  §  ttv, 

=  1,   if  7-  =  0, 

z=  -  i  +  1  V^,  if  r  =  1,> 

=  -i-^V^  if  r  =  2; 

therefore  the  roots  are  1,  l{-  1  +  V^),  ^{-l  -  V^). 

[Remaek.  — The  abridged  notation  of  Hamilton  bears  a  close  analogy  to 
the  corresponding  expressions  deduced  from  Euler's  formulas.  These  for 
mulas  are 

cos  0  =  i(eVrr  +  e-*^/^),  (a) 

V^  sin  9  =  i(e9v/^_  e" V~i),  (b) 

in  which  e  is  the  base  of  the  Napierian  system  of  logarithms.  Adding 
gives 

cos  0  +  /^  sin  0  =  /^ "^  (c) 

Hamilton's  notation  is 

co8e+  fC4sin9  =  c'.  (d) 


286  QUATERNIONS.  [38a 

Equation  (c)  is  an  equality  of  values,  while  {d)  is  a  notation. 
In  (c)  let  e  =  {An  +  ^)Tt,  in  which  n  is  one  of  the  series  0,  1,  2,  3,  etc., 
and  it  becomes 

.  \  log  V^^  =  (4w.  -t-  i)  7C  V^, 

hence  the  log  of  V—1  has,  analytically,  an  infinite  number  of  imaginary 
values.     If  n  =  0,  we  have 

log  V^  =  ^n  V^. 

This  V—1  is  not  a  unit-vector,  but  the  imaginary  of  algebra.     (See 
Art.  355.)] 

APPLICATIONS. 

380.  Triedrals. — Let  the  diedral  angle  at  a  be  A,  at 
Py  B,  at  y,  G,  and  the  facial  angles  op- 
posite these  be  a,  b,  c,  respectively.  Also 
refer  to  the  plane  AOB  as  c,  and  simi- 
larly for  the  others.  At  the  initial  point 
0  erect  unit-vectors  perpendicularly  to 
the  respective  planes 

Fig.  259.  ^    I    i,  c     I    ^  c     '    ^ 

and  they  will  be  the  operators  for  the  respective  vectors,  and 
also  form  the  edges  of  another  triedral,  the  diedral  angles 
of  which  will  be  the  supplements  of  the  diedrals  of  the 
given  triedral.  If  an  operator  be  drawn  from  any  other 
point  of  the  plane  than  at  the  vertex,  it  will  either  pass 
through  the  triedral,  or  else  lie  entirely  without  it.  Con- 
sider those  as  positive  which  lie  without,  and  the  opposite 
as  negative,  and  that  the  rotation  from  the  vector  of  the  de- 
nominator to  the  vector  of  the  numerator  is  positive.  Then 
we  have 

^  -L  -7. 

—  =  coso—  f  sino, 

y 

-5  =  cose  +  fiSinc, 

ft 

-  =  cos  a  +  f  o  sm  a, 

Y 
and  equation  (3),  Article  378,  becomes 

cos  &  —  f  sin  h  —  (cos  c  +  fj  sin  c)  (cos  a  +  fa  sin  o),     (1) 


880.]  APPLICATIONS.  287 

the  second  member  of  which  has  already  been  developed  in 
Example  9,  Article  368.  Taking  the  scalar  part,  observing 
that  the  angle  between  s^  and  tg  is  180'—^,  we  have 

cos  h  —  cos  a  cose  +  sin  a  sin  h  cos  B,  (2) 

which  is  one  of  the  fundamental  equations  of  spherical  trigonom- 
etry. 

If  B  —  90'',  we  find  one  of  the  formulas  for  right-angled 
spherical  triangles. 

Taking  the  vector  parts  and  dividing  by  fj,  we  have 

sin  &=cosasinc-r  —  sinacosc-1 —  sinasincsin  B.    (3) 

Applying  the  preceding  rules  to  the  triedral  6  f i  f2 ,  we 
have 

-  =  cos  (180°- J)  -  «sin  (180"-  A) 


=  —  cos  A  —  a  sin  A, 
i?  =  cos  (180°  -B)  +  ft  sin  (180°  -  B) 

=  —  cos  B  +  ft  sin  B, 
4  =  cos90°  +  f3sin90°  =  f3, 

Substituting  in  (3)  and  taking  the  scalars,  gives 

sin  b  cos  A  —  cos  a  sine  —  sin  a  sin  c  cos  B,        (4) 
which  is  another  important  fcyrmula  in  spherical  trigonometry. 
Taking  the  vectors  gives,  after  dividing  by  ft, 

5sin&sin  J=  sinacoscsin^  +-^  sin  a  sin  c  sin  5, 
ft  ft 

in  which  substitute  the  values  of  ^  and  -^  given  above,  and 
it  becomes 

sin  &  cos  c  sin -4  +  fi  sin  6  sine  sin  ^  =  sin  a  coscsin^  4- 

fi  sin  a  sine  sin  i?. 


288  QUATERNIONS.  [381,383. 

The  scalars  and  the  vectors  of  this  equation  give  two 
identical  equations,  either  of  which  is 

sin  h  sin  A  =  sina  sin  B, 
or  sin  h  :  sin  a  : :  sin  B  :  sin  A, 

that  is, —  The  sines  of  the  angles  of  a  spherical  triangle  are  prO" 
portional  to  the  sines  of  the  opposite  sides. 

Other  relations  may  be  found  by  making  other  combina- 
tions of  the  vector  axes  in  equation  (3).* 

381.  Relation  between  the  vector  to  a  point  and  the  Cartesian 
coordinates  to  the  same  point. 

Let  a,  /?,  y,  be  unit- vectors  along  the  axes  x,  y,  z,  respec- 
tively, and  p  the  vector  from  the  origin  to  the  point ;  then 

p  =  xa  +  y/3  +  zy. 

If  the  axes  are  rectangular,  then 

p  =  ix  +jy  +  kz. 

Multiply  through  by  i,  and  we  have 

Sip    =      —     Xy 

and  similarly,        Sjp  —  —y,     Skp  =  —  2, 

.'.  p^  =  x^  +  y^  +z\ 

=  {SipY  +  {Sjpy  +  (Skpy. 

Let  r  be  the  scalar  of  p,  then  Sip  =  —r  cos  X,  etc.,  and 
we  have 

r^  =  r-  cos'  X  +  7^  cos^  Y+  r'^  cos-  Z, 

or  1  =  cos-  X  +  cos^  Y  +  cos-  Z, 

which  is  equation  (3),  Article  198. 

382.  Conjugate  quaternions  in  reference  to  each 
other  are  such  as  are  equal  in  all  respects,  except  that  their 
angles  have  contrary  signs.  Thus,  in  unit-vectors,  if  q  be 
the  given  quaternion,  and  Kq  is  conjugate,  we  have 

*  Other  relations  are  given  in  Hamilton's  Lectures,  p.  537. 


383.]  APPLICATIONS.  283 

q=  —  cos  6  +  e sin ^, 
Kq  =  —  cos  d  —  a  sin  ^. 
Generally,  if  q  =  Sq  +  Vq, 

then  Kq=  Sq-  Vq, 

and  q^q=(Sqy  +  {TVq)% 

since,  (359),  (Fc/)^  =  -  (2^r^)^. 


10 


CHAPTEE  HL 


OP    THE    LINE,    PLANE,    SPHERE,    AND    CONIC 
SECTIONS. 

Right  Line. 

383.  Vector  Equation  to  a  Right  Line.  — The  posi- 
tion of  a  right  line  may  be  determined  in  several  different 
ways. 

1.  A  right  line  is  determined  when  two  points  of  it  are 

known.  Let  A  and  B  be  two 
points  of  the  line,  0  the  origin 
of  vectors,  a  and  ft  the  known 
vectors,  C  any  point  of  the  line, 
and  p  the  vector  to  C. 

Then  since  AC  is  parallel 
to  A  B,  (in  this  case  coinciding 
with  it),  we  have  (327)  AC  = 
xAB.    The  triangles  OAB  and 
0J(7give 

OA  +  AB  =  OB, 

OA  +  AG  =  OC; 

or  a+  AB  =  /?, 

a  +  xAB  =  p. 

Eliminating  AB  gives 

p  =  a  +x{/3  —a), 


FIG.S60. 


(1) 


(2) 


which  is  the  required  equation. 

2.  A  right  line  is  also  known  when  the  length  and  posi- 
tion of  a  perpendicular  to  the  line  are  known. 

290 


383.]  BIGHT  LINE.  291 

Let  AP  be  the  line,  0  the  initial  point  of  vec-     a    b 
tors,  OA  =  6  the  vector  perpendicular  to  AP,  P    ^ 


any  point  of  the  line,  OP  =  p,  and  y  a  unit-vector      [/^ 
to  which  the  line  is  parallel,  and  AB  =  y:  then       o 

Fia.  261. 

AP  =  xy, 
and  p=  OA  +  AP 

=  S  +  xy.  (3) 

This  may  be  put  in  another  form.  Multiplying  both 
sides  into  6  gives 

pd=6^  +  xyS.  (4) 

But  xy  and  d  being  mutually  perpendicular,  we  have 
xSyd  =  0,  (Art.  368,  Ex.  12) ;  and  if  the  tensor  oi  d  he  d  and 
of  p  be  p,  (4)  becomes 

Spd  =  -cP,  (359),  (5) 

or  —  pc?cos  (AOP)  —  —  d\ 

or  p  cos  cp  =  d,  (6) 

which  is  a  polar  equation  to  a  line,  (37,  7°).  In  (5)  y  has 
disappeared,  hence  the  line  may  be  positive  in  either  direc- 
tion. The  line  is  in  the  plane  of  d  and  p,  which  fact  may  be 
expressed  by  the  equation 

Sep  =  0.  (7) 

(Obs.— Dividing  (3)  by  S,  and  taking  the  scalars,  (367,  (7))  gives 

P  -I 

^cos  O)  =1, 
a 

or  p  cos  g>  =  d, 

as  before.) 

3.  A  right  line  is  given  when  its  direction  and  any  point 

of  it  are  known.  b 

Let  B  be  the  given  point,  P  any  point,  y        0y 

a  unit  vector  parallel  to  the  line ; 

0 

then  BP  =  xy,  j.^^  262. 

and  p=  OB  +  BP 

=  fi  +  xy,  (8) 

which  is  the  required  equation. 

[Remark. — Of  the  equations  (2),  (5),  and  (8),  sometimes  one  is  better 
adapted  to  the  solution  of  a  particular  example  than  either  of  the  others.] 


292  QUATERNIONS. 


EXAMPLES. 


1.  To  find  the  equation  to  a  line  which  shall  pass  through  a  given 
point. 

Let  /?  be  the  vector  to  the  point ;  then  equation  (8)  will  be  the  re- 
quired equation,  in  which  p  and  y  are  both  unknown ;  hence  an  infinite 
number  of  lines  may  be  drawn  through  a  point. 

2.  To  find  the  equation  of  a  line  which  shall  pass  through  a  point 
and  be  perpendicular  to  a  given  line. 

Let  P  be  the  vector  to  the  given  point,  and  5  the  vector  to  which  the 
line  is  perpendicular. 

Draw  a  vector  y  through  the  given  point  perpendicular  to  5,  then 

8yd  =  0. 
Let  /3  be  a  vector  to  any  point  of  y,  then  (Eq.  (8)), 
p=  /J  +  xy, 
will  be  the  required  equation.     We  also  have 

pS  =  fid  +  xy8, 
or  8pS  =  S/Jd, 

since  SyS  =  0,  Since  y  has  disappeared  from  this  equation  it  appears 
that  an  infinite  number  of  lines  may  pass  through  a  point  jierpendicular  to 
a  line.  But  if  the  lines  S  and  y  are  embraced  by  one  plane,  we  have  the 
additional  condition 

Sep  =  0,    . 

in  which  case  only  one  line  can  be  drawn  perpendicular  to  the  given  line. 

3.  Find  the  equation  to  a  line  which  shall  pass  through  two  points. 
(This  is  equation  (3).) 

4.  Find  the  distance  between  two  given  points. 

Let  p  and  p  be  vectors  to  the  given  points,  and  AB  the  distance  be- 
tween them ;  then 

AB  =  p  —  p. 

For  the  remainder  of  the  solution,  consult  Article  374. 

5.  To  find  the  length  of  a  perpendicular  from  a  given  point  to  a  given 
line. 

Let  the  line  be  given  by  equation  (8),  the  initial  point  being  at  the 
given  point.  When  p  is  perpendicular  to  BP,  let  its  value  be  S,  and  w« 
have 

S  z=z  /3  +  xy, 

or  d^  =  d/3  +  xSy, 

or  Sd''  =  SSfi  +  0, 

or  -d'  =  S8/3; 


384]  THE  PLANE.  293 

U6  being  a  unit-vector,  d  the  length  of  the  perpendicular,  and   there- 
fore d.Ud  =  S. 

6.  Find  the  equation  to  a  line  perpendicular  to  each  of  two  given 
lines. 

TJie  Plane. 

384.  Equation  to  the  Plane. — If  the  plane  be  given 
by  the  condition  that  it  shall  pass  through  a     ^v    b      p 
point  and  be  perpendicular  to  a  liue,  its  essential   ^ 
equation  will  be  the  same  as  equation  (5)  of  the 
preceding    Article,    subjected   to   the   condition    o 
that  p  shall  not  be  confined  to  a  plane ;  hence       ^'^  ^^■ 
the  equations  are 

Spd  =  —  d-  =  C  {a  constant),  (1) 

and  Ssp  =  an  indeterminate  quantity. 

Equation  (1)  may  be  written 

^^  =  1.  (2) 

o 

If  the  origin  be  in  the  plane,  cos  0  of  equation  (5),  Arti- 
cle 367,  will  be  zero,  and  the  equation  becomes 

SpS^O; 

in  which  case  p  will  be  independent  of  6. 

EXAMPLES. 

1.  To  find  the  equation  of  a  plane  which  shall  pass  through  three 
given  points. 

Let  a,  /J,  y.  be  the  given  vectors,  and  p  the  variable  vector.  Then 
p  —  a  is  a  vector  in  the  given  plane,  and  similarly,  a  —  (S,  fi  —  y\  hence, 
{Art.  368,  Ex.  10,  Eq.  (3)], 

5(p-a)(a-/J)(yS-r)  =  0,  (1) 

which  may  be  reduced  to 

Sp  (  Vafi  ■¥  Vfiy  +  Vya)  -  Sa/3y  =  0,  (2) 

which  is  the  equation  sought. 
[For  expanding  we  have 

8[  pafi  -  pfi^  -  a*li  -t-  a/5»  -  pay  -f  p(iy  +  a'^y  -  a/3y]  =  0. 
But,  8p/S*  =  0,  for  8p/3»  =  Sp  (-  1)  =  0,  since  it  has  no  scalar  part; 


294  QUATERNIONS.  [384 

(Art.  368,  Ex.  3),  and,  similarly,  for  other  terms  containing  squares.  Hence 
we  have 

8{pafi  —  pay  +  pf^y  —  ocfdy]  =  0, 

or,  (357),  8  [pa/3  +  pya  +  pfJy]  —  Safty  =  0. 

But  Spaft  =  -  raJ  sin  (a,  6)  sin  (r,  a6),  (Art.  368,  Ex.  10,  Eq.  (1)). 
We  have        Va^  =  a&  sin  {a,  b).s,  (Art.  367,  Eq.  9)), 
and  8p  Fa/J  =  Sab  sin  (a,  6)  pe, 

also  pB  =  r(—  cos  (r,  e)  +  e'  siQ(r,  c) ; 

.  *.  8p  Va/J  =  Srab  [—  sin  (a,  b)  cos  {r,  e)  +  e'  sin  (o,  6)  8in(r,  «)] 
=  —  rob  sin  (a,  6)  cos  (r,  e)  * 

=  —  ra&  sin  {a,  b)  sin  (/•,  ab) ; 
.:  8pVafi=  SpafS, 
and  similarly  for  the  others.] 

2.  A  plane  cuts  off  a  pyramid  from  the  three  rectangular  coordinate 
planes ;  i-equired  the  locus  of  the  foot  of  the  perpendicular  from  the  ori- 
gin upon  the  plane  when  the  volume  is  constant. 

Let  the  three  rectangular  unit- vectors  be  i,  j,  A;;  and  the  vectors  at, 
Ij,  ck\  and  S  the  perpendicular  from  the  origin  upon  the  plane;  then,  as 
in  the  preceding  example,  we  have 

S  —ai,         S  —  bj,         8  —  cJc, 
three  vectors  in  the  plane. 

But  since  d  and  these  vectors  are  perpendicular  to  each  other,  we 
also  have,  (Art.  368,  Ex.  12), 

SS(S-ai)=0,         /S5  (d -&;■)=  0,         SS{S-ck)=0', 
or  -d^  =  aSSi,         -  d^  =  hSSj,         -  d^  =  cSSk. 

Multiplying  together  gives 

-d^  =abe  S8i  SSj  SSh, 

which  will  give  the  perpendicular  upon  the  plane  for  any  given  position 
of  6.  The  product  abc  is,  according  to  the  problem,  constant.  Let 
—  d*  =  5*  be  variable,  and  we  have,  from  Cartesian  coordinates, 

8^  =x^  +y^  -|-s^ 

SSi  =  —  8  cos  (S,i)  =  —8  cos  {8,x), 

cos  (5,  *)  =  ^ ; 


and  we  have 


SSi  =  —X, 


a  surface  of  the  sixth  order. 


385.]  TUE  CIRCLE.  295 

A  solution  miglit  be  effected  by  operating  upon  equation  (2)  of  tlie 
preceding  example,  but  it  is  not  so  direct  as  the  above. 

The  Circle. 

385.  Equation  to  the  Circle. — Let  G  be  the  centre  of 

the  circle,  0  the  origin  of  vectors,  CP       ^ ~xp 

—  r  —  the  radius,  OC  =  y  =  the  vector 
to  the  centre,  P  any  point  whose  vector 
is  OP  =  p,  and  a  —  vector  CP. 

The  triangle  OCP  gives  p^^  ^^ 

p=y  +  a,  (1) 

which  must  be  subjected  to  the  condition  that  CP  =  r  =  a 
constant.     We  find 

{p  -yf=a^^-  r\  (2) 

which  is  one  form  of  the  required  equation.  Squaring  the 
first  member,  making  OC  —  c,  this  becomes,  (Art.  368,  Ex.  6), 

p'-'lSpy=(?-r\  (3) 

If  the  origin  0  be  upon  the  circumference,  we  have  0  G 
=  r  and  (3)  becomes  ^^  ^ x^p 

p'  -  '2Spy  =  0.  (4) 

Substituting  the  value  of  Spy  =  —  p'r  cos  6, 
(4)  becomes  ^^i^. 

p  —  2r  cos  ^  =  0, 

which  is  the  polar  equation  to  the  circle,  the  origin  being  on 
the  circumference,  and  the  diameter  the  initial  line.  HOG 
=  ^y,  (4)  becomes 

p^-Spy=0.  (5) 

If  the  origin  be  at  the  centre,  the  length  of  y  wiU  be  zero, 
c  =  0  and  (2)  becomes 

//  =  a^  =  —  r^,  (6) 

that  is  the  radius  vector  is  constant.  If  —p'^he  the  scalar  of 
p^  the  equation  becomes 

p'^=  A 


296  qUATEBNIONS.  [386. 

EXAMPLES. 

1.  An  angle  inscribed  in  a  semicircumf erence  is  a  right  angle, 
p  Equation  (4)  may  be  written 

Sp{p-2r)  =  Q\ 

hence  the  vectors  p  and  p  —  2y  make  a  right  angle, 
(Art.  368,  Ex.  12).  But  'iy  z=  "iOC  =  OA,  a  diameter, 
and  the  vector  equation 

OA-^  AP^  OP, 
gives  AP  =  p  —  2y; 

hence  OP  and  PA  make  a  right  angle. 

2.  A  line  is  drawn  at  random  through  a  fixed  point  A  and  a  perpen- 
dicular is  let  fall  upon  it  from  another  fixed  point  P;  to  find  the  locus 
of  the  intersection. 

Let  a  be  the  vector  joining  A  and  P,  ^  a  unit-vector  through  P  and 
p  the  variable  vector  through  J.,  then  will  the  equation  of  the  line  be, 
(382,  Eq.  (8)), 

p  =  a  +  «/?. 

Let  8  be  the  perpendicular  fi-om  A,  then  we  have 

8  =  a  +  yfS, 
and  52  =  5a  +  y5/J; 

.-.  (5«  =  SSa, 

for  88 fi  =  0.     Hence,  (Eq.  (5) ),  the  locus  is  a  circle  whose  diameter  is 
a  =  r. 

386.  Equation  of  a  Tangent  Line  to  a  Circle. — Take 
the  origin  at  the  centre  of  the  circle,  a  a  radius  vector  to 
T  the  point  of  tangency,  r  a  vector  from  the  centre  to  any 
point  A  of  the  tangent,  then  we  have 

AT=  r-a. 

But  oc  and  A  T  make  a  right  angle,  hence 

Sa{T-a)  =  Q,  (1) 

or  -7^z=a^=  Sat,  (2) 

or  l  =  iS'-,  (3) 

a 

is  the  required  equation. 


887.]  THE  SPHERE.  297 

EXAMPLE. 

The  square  of  a  tangent  to  a  circle  equals  the  product  of  the  whole 
secant  into  its  external  segment. 

Let  TA  be  a  tangent,  AD  the  secant,  vector 
CT  =  a,  CA=:  r,  TA  ^  y,  and  lines  CT  ^  CB 
^CD  =  r,  CA  =  s,  TA  =  t. 


Then 

T  -a  +  y, 

squaring, 

r^  =r  a2  +  2Say  +  y^. 

But 

2 Say  =  0,  (Eq.  (1)  above) ; 

.-.   y-^  =r3-a2, 

or  as  lines 

_  e^  =  .-  s  2  +  r^, 

or 

t^  =  (s  +  r)  {s  —  r) 

=  AH .  AB. 

Fig.  267. 


The  Sphere. 

387.  Equations  of  the  Sphere. — If  the  lines  in  Fig. 
264,  Article  385,  are  not  restricted  to  one  plane,  equations 
(3),  (4),  (6),  will  be  the  equations  of  the  sphere,  and  equa- 
tions (1),  (2),  (3),  of  Article  386  will  be  the  equation  of  a 
tangent  plane. 

EXAMPLES. 

1.  Every  section  of  a  sphere  made  by  a  plane  is  a  circle. 

Let  5  be  a  vector  perpendicular  to  the  plane,  d  the  scalar  of  5,  r  the 
radius  of  the  sphere,  p  the  vector  to  the  line  of  intersection  of  the  plane 
and  sphere,  fi  a  vector  in  the  i:)lane  connecting  8  and  p.  Then  we 
have  the  vector  equation 

/3  =  5  +  /?, 

squaring,  p^  =  S^  +  2SSj3  +  /P  ; 

or  as  lines,  _  -^8  =  _  ^^a  _  2d/3  cos  90°  -  b^ ; 

.-,  S/3^  =  d^  —  r^  =  a  constant; 

hence,  (Eq.  (6),  Art.  385),  the  line  of  intersection  is  a  circle.     It  is  real 
for  d  <r.     (Observe  that  Sft'^  is  negative.) 

2.  Find  the  curve  of  intersection  of  two  spheres. 

Take  the  origin  anywhere ;  then  equation  (3),  Article  385,  gives 
p«  -  28yp  =  C, 
p8  -  2Sy  >  =  G\ 
subtracting,  28{y'  —y)p—  C—  C  =  a  constant, 


298 


QUATERNIONS. 


■which  is  the  equation  of  a  plane,  hence,  from  example  1,  we  see  that  the 
intersection  is  a  circle. 

The  Ellipse. 

388.  Equations  to  the  Ellipse. — 1.  Let  the  origin  be  at 
the  centre,  p  a  vector  to  any  point 
P  in  the  curve,  y  a  vector  to  the 
focus  F,  the  length  of  CF  being  c, 

a=  CA,b=  CB,e=  -. 
a 

We  have 

vector  FP  =  p—  y, 
F'P  =  p  +  y, 
and  by  the  definition  of  an  ellipse 

T{p-y)+  T{p  +  y)  =  2a, 
or  V-  (P  -  yy  +  V-{P  +  yr  =  2a; 


,'.  V-  {P  -  yy  =  2a  -  V-  {p+yY . 
Squaring  gives 

-  (p2-  2Spy  +  f-)  =  4:0?-  4a  V-  {p  +  yf  -  {p'+2Spy  +  y^) ; 


or         .  Spy  =ar  —  a  V—  {p  +  yf . 

Transposing  a^,  squaring  and  reducing  gives 

a'/f+{Spyf=-a'-aY 
=  —  a*  +  aV 
=  -  a*  +  aV 

which  is  the  required  equation. 

Substituting       (SpyY  =  {—  py  cos  Oy 

=  pV  cos^  6 

(observing  that  p  is  used  to  represent  both  vector  and  scalar 
BO  that  p^  as  vector  =  —  p^  as  a  line),  gives 

ah 


(1) 


Va^  —  c"  COS"  0 

ab 
Va^  sin^  6  +  b"  cos"  0, 


(2) 


389,  890.] 


THE  PARABOLA. 


299 


which  is  the  polar  equation  to  the  ellipse,  the  pole  being  at 
the  centre,  Article  174. 

2.  Let  t,y,be  unit- vectors  along  the  axes  then  will  the 
equation  be 

p  =  xi  +  yj,  (3) 

in  which  x  and  y  are  related  by  the  equation 

3.  Let  a,  /?,  be  unit-vectors  along  conjugate  diameters. 
Then  the  equation  becomes 

/J  =  xcx  +  yfi,  (4) 

in  which  x  and  y  are  related  by  the  equation 

2         &'  '  /     ■  o  -n 


TJie  Hyperbola. 

389.  The  Equations  to  the  Hyperbola  are   the  same 
as  for  the  ellipse  excepting  that  e  exceeds  unity. 

Tlie  Parabola. 

390.  Equation  to  the  Parabola. — Let  the  origin  be  at 
the  focus  F,   FP=  p,  FA  =  r,    OG  =  /i,    q, 
the  directrix;    then  FO  -  2/,    PG  —  xy,    g 
and  as  lines  FP -=  PG,  FA  :=  AO  ^  p. 

We  have  the  vector  equation 

FP  +  PG^FO  +  OG, 

or  p  +  xy  =  2y  +  /3 ; 

then  py  +  xy'^  =  2/^  +  ^y, 

and  Spy  +  Sxy'  =  2Sy\ 

for  /S'/?x  =1  0,  since  /3  and  ^^  are  mutually  perpendicular ; 
then  Spy  —  xp^  —  —  2p^ ; 

Spy  +  'If 


Fig.  269. 


(1) 


.*.  X  = 


f 


(2) 


300  QUATERNIONS.  [390. 

From  the  property  of  the  parabola,  we  have 

FP  =  PO 

or  Tp  =  xTy  ; 

hence  as  vectors,  —  f^  =  —  x^y^,  (3) 

and  as  lines  r  —  xp,  (4) 

where  r  is  a  variable  line.     Eliminating  x  between  (2)  and 
(4)  gives 

pr  —  2p-  +  Spy  (5) 

which  is  the  required  equation.     Substituting  Spy  =  —  rp 
cos  6,  gives 

pr  =  2p^  —  rp  cos  0, 

.  r- ^^ (6) 

••"^-l  +  cos^'  ^^ 

which  is  the  polar  equation  to  the  parabola,  the  parameter 
here  being  4p,  (Art.  168). 

(Since  y'^  has  no  vector  part,  the  value  will  be  the  same  if  ^S'  be  omitted 
before  y^  and  equation  (1)  becomes 

Spy  +  xy^  =  2y^  ; 
2y^  -  Spy 

which  in  (8)  gives 

y^p^  =  {2y^  -  SpyY,  (7) 

which  is  the  required  equation  ia  terms  of  vectors  only.     Substituting  in 
it  the  values 

y^  =  —  p^,  p2  _  _  ^2^  gpy  _  _  j,p  gQg  g^ 

we  have  r  =  2p  —  r  cos  9 ; 

•  • ''  ~  1  4-  cos  e ' 

which  is  the  same  as  equation  (6)). 

By  means  of  these  equations  all  the  properties  of  the 
conic  sections  may  be  discussed. 


PART   III. 


MODERX    GEOMETRY. 


MODERN    GEOMETRY. 


CHAPTEE  I. 


391.  Modern  Geometry  includes  all  the  systems  of  ge- 
ometry which  have  arisen  since  the  Cartesian  system  was 
recognized ;  but  the  term  is  generally  restricted  to  the  sys- 
tem called  Trilinear  Coordinates. 

392.  Definition  of  a  Locus  extended. — Any  system 
of  lines  which  make  known  a  curve,  may  be  called  a  system 
of  coordinates ;  hence  the  equation  to  a  locus  is  an  equation 
which  expresses  the  relation  betiveen  any  system  of  lines  ivhich  de- 
termine a  locus. 

Tangential  System. 

393.  Conceive  that  tangents  are  drawn  at  all  the  points 
of  any  curve,  and  that  the  curve  is  then  removed.  It  will 
be  easy  to  retrace  the  locus  by  drawing  a  curve  which  will 
be  tangent  to  the  consecutive  lines.  Such  a  locus  is  said  to 
be  an  envelope  of  the  system  of  right  lines,  and  any  equation 
which  will  determine  the  consecutive  positions  of  the  tan- 
gents will,  in  this  way,  determine  the  locus. 

394.  The  four  cusp  Hypocycloid. — To  illustrate,  it  is 
known  that  the  tangent  to  the  four  cusp  hypocycloid  between 
the  rectangular  axes  is  constant.  Therefore  if  a  right  line, 
equal  in  length  to  the  radius  of  the  directrix,  be  made  to 
slide  on  the  axes  CB  and  CD  keeping  both  ends  in  contact 
with  these  lines,  they  will  in  all  positions  be  tangent  to  the 


304 


MODERN  GEOMETRY. 


[395. 


hjpocycloid  DB.  And  if  an 
indefinite  number  of  lines  be 
drawn  upon  the  paper,  coin- 
ciding with  the  several  po- 
sitions of  the  tangent  line, 
the  hypocycloid  may  be 
B  drawn  by  making  it  tangent 
to  the  successive  lines. 

To  find  its  equation  in  terms  of  the  tangential  intercuts. 
Let  EF  be  one  of  the  tangents ;  then  will  CE  be  the  in- 
tercept on  the  axis  of  x,  and  CF  on  the  axis  of  y.     Also  let 

CE  =  —  Xt,         CF  =  yt ,         EF  =  1,  a  constant, 

then  we  have        a^  -\-  y\  =  J?^ 

for  the  required  equation. 

395.  Equation  to  the  Ellipse  in  terms  of  tangential  irir- 
terc&pts.     The  equation  of  the  tangent  is,  (Art.  100), 


Wx  X  -\-a-y'y 

=  a 

%\ 

and  the  intercepts  are, 

(104), 

CT-- 

-  x"" 

x'  -. 

a- 

CT- 

w 
^y"" 

y'-- 

^yr 

Fig.  371. 

•  and  since  the  point  x'y'  is  on  the  curve,  we  have 
in  which  substitute  the  values  of  x  and  y\  and  we  have 

^+-^=1 

^.      yl      ' 

which  is  the  required  equation. 
1 


Let 


=S,  and  —  =  w, 

^t  yt 


and  the  equation  becomes 


396,  397.  J 


TASGEXTIAL   SYSTEM. 


305 


396.  Equation  to  a  point  in  terms  of  taTigential  inter- 
cepts.— Every  Hue  passing  througli  a  point  is  (by  an  exten- 
sion of  the  definition)  tangent  to  y 
the  point.  Let  P  be  the  point 
and  KH  a  line  passing  through  it ; 

OA=a,AP=h=0B,    1-1 


OH      '  OK 


=  Tf. 


or 


We  have 

HO  :  PB  ::  OK 

KB, 

1              11 

-^  :  a  ::  ~   :  -- 

-h; 

Fig.  272. 


(1) 


.'.  a^  +  hi]  =  1, 

which  is  the  required  equation.  By  means  of  this  equation 
any  number  of  lines  may  be  constructed,  all  of  which  will 
pass  through  the  point  P,  and  thus  the  point  becomes  deter- 
mined. 

If  the  line  HK  be  given  in  terms  of  the  perpendicular, 
OQ  =p,  and  the  angle  QOX  =  cp,  we  have 

^  _   cos  cp  _  sin  qj 

c,  —  ,  t]  —  , 

i>  p 

and  the  equation  becomes 

a  cos  fp  -\r  h  sin  <p  =  p.  (2) 

Still  further,  if  OP  =  c,  and  POA  =  a,  equation  (2)  be- 
comes 

p  =  c  cos  {(p  —  i^),  (3) 

which  latter  equation  is  called  the  tangential  polar  coordinates 
of  the  locus.  (See  Quarterly  Journal  of  Pure  and  Applied 
MatJiematics,  Vol.  I.,  p.  210.) 

397.  Tangential  equation  to  a  Right  Line. — Only  one 
right  line  can  be  tangent  to  a  right  line,  hence  its  intercepts 
will  be  constant,  and  we  have 

^  =  a,        V  =l>f 
for  the  required  equations. 

We  see  that,  in  this  system,  a  point  is  given  by  one  equa- 
tion, while  a  line  requires  two  equations. 
20 


306  MODERN  GEOMETRY.  [39a 

EXAMPLES. 

1.  Find  the  equation  to  a  circle  in  terms  of  tangential  intercepts. 

2.  Find  the  equation  to  a  hyperbola. 

3.  Find  the  equation  to  a  2)arabola. 

4.  Prove  tliat  tlie  distance  between  the  points  a|  +  St?  =  1,  and 
a^  +  o'lj  =  1,  is  V{a'  —  «)«  +  (b'—  6)«. 

5.  Prove  that^j  -d  +  n  cos  q)  represents  a  circle,  d  and  n  being  fixed 
numbers,  and  determine  its  radius. 

398.  A  line  may  be  determined  by  the  perpendicu- 
lar distances  to  it  from 
three  fixed  points.  The 
fixed  points  will  be  the 
vertices  of  a  triangle. 
Let  A,  B,  C,  be  the 
points,  then  will  the 
perpendiculars  <r,  fi,  y, 
determine  the  line  RP. 
The  perpendiculars 
will  have  the  same  sign, 
if  the  required  line 
does  not  pass  between  the  points  from  which  the  perpendic- 
ulars are  drawn,  otherwise  they  will  have  contrary  signs. 
Thus,  in  the  figure,  a  and  y  will  have  the  same  signs,  both 
plus  or  both  minus ;  but  a  and  ft,  and  ft  and  y  will  have  con- 
trary signs.     If  the  required  line  bisects  AB,  we  have 

a=  —  ft,  or  a+ft  =0, 

which  may  be  considered  as  the  equation  of  the  point  of 
bisection  of  AB.  If  the  line  BP  were  parallel  to  AC,  we 
would  have 

a=y, 

and  if  a  and  ft  have  contrary  signs  it  will  cut  the  other  two 
sides  of  the  triangle,  but  otherwise  it  will  lie  either  entirely 
above  or  entirely  below  the  triangle. 

Any  curve  may  be  determined  by  this  system  by  making 
it  the  envelope  of  a  system  of  lines  determined  according  to 
a  proper  law.  This  system  is  generally  considered  as  a 
modification  of  the  trilinear  system. 


Fia.  273. 


399-401.] 


TRILINEAB  SYSTEM. 


307 


Fig  274. 


Trilinear  System. 

399.  Wlien  a  locus  is  determined  by  tliree  perpendicu- 
lars from  each  point  upon  the  three  sides  respectively  of  a 
given  triangle,  the  system  is 
called  trilinear.  The  given 
triangle  ABC  is  called  the  tri- 
angle of  reference.  Let  P  be 
any  point ;  from  it  let  fall  the 
perpendiculars  PE  upon  CA, 
Pi)  upon  CB,  PF upon  AB; 
and  let  the  perpendicular  upon 
the  side  opposite  the  angle 
A  be  (Y,  opposite  B  be  fi,  and  opposite  C,  y.  Then,  conversely, 
if  a,  fi,  y,  are  given,  the  point  P  will  be  determined. 

We  have  previously  seen,  (Art.  8),  that  a  point  may  be 
determined  by  its  reference  to  tico  axes,  and  hence  it  appears 
that  ^}lre^  are  unnecessary  ;  but  it  will  soon  appear  that  when 
three  are  used  the  expressions  have  a  certain  symmetry 
which  they  do  not  have  when  only  two  are  used ;  and  also 
that  the  angle  between  the  two  axes  does  not  aj^pear. 

400.  Signs  of  the  Perpendiculars. — Those  perpendic- 
ulars will  be  positive  which  lie  on  the  same  side  of  the  line 
of  reference  as  the  corresponding  angle.  Thus,  in  Fig.  274, 
fi  and  B  are  on  the  same  side  of  A  C,  and  hence  /^  will  be 
positive,  but  in  Fig.  275  Ph,  or  /?,  and  B  are 
on  opposite  sides  of  AC,  and  hence  /:^in  this 
case  will  be  negative. 

When  the  point  is  within  the  triangle  of 
reference,  all  the  perpendiculars  will  be  posi- 
tive, but  when  the  point  is  without  one  or  tico 
will  be  negative,  but  in  no  case  will  all  three  be  negative. 

401.  Relation  of  the  Coordinates. — Let  ^  be  the  area 
of  the  triangle  of  reference  A  BC,  a  =  CB,  the  side  of  the 
triangle  opposite  A,  b  =  AC,  c  =  AB,  and  conceive  lines  to 
be  drawn  from  P  to  the  vertices  of  the  triangle  ;  then  will 

aoc  =  twice  the  area  of  the  triangle  PBC, 

J^  _        u  «         «  «        u  u  pQj^^ 

cy  =     "        "      "      "     "        *'        PBA : 


Pio.  275. 


aa  -t  bj3  +  cy  =  2  J. 


(1) 


308  MODERN  GEOMETRY.  [401 

If  the  point  is  without  the  triangle,  one  or  more  of  the 
coordinates  a,  /i,  y,  will  be  negative,  and  2^  will  be  the  dif- 
ference of  the  terms  in  equation  (1). 

Let  p  be  the  radius  of  the  circle  circumscribing  the  tri- 
angle ABC;  then  we  have 

s5r2  ~  sm5  ~  S77  ~  ^'  ^  ^ 

which  combined  with  (1),  gives 

a  sin  A  4-  ySsin  B  -\-  y  sin  C=  —  =  S,  (say).     (3) 

Equations  (1)  and  (3)  are  useful  in  transforming  other 
equations  so  as  to  make  them  homogeneous.  For  from  (1) 
we  have 

aa  +  b/3  +  cy  _^ 
2^  ~    ' 

and  hence  we  may  multiply  any  term  of  an  equation  by  the 

fraction ^-j ^ ,  thus  raising  by  unity  the  order  of  the 

term  without  changing  its  value.     Thus,  to  render  homoge- 
neous the  equation 

a^  +  3yS2  +  2y  =  l, 

we  raise  each  term  to  the  third  degree  as  follows : 

«  ,  QA2««  +  b/3  +  cy  ^   „    faa  +  b/3  +  cyy_{aa  +  b^  +cyy 
"^  +  "^^  2^  +  ^^  V         2^         /  "\         2J        J ' 

which  may  be  expanded  and  reduced. 

EXAMPLES. 

1.  Show  that  the  coordinates  of  the  middle  point  of  BG  are 

0  c 

2.  Show  that  the  coordinates  of  the  angle  A  are 

3.  The  coordinates  of  the  foot  of  the  perpendicular  from  A  upon  BO 
are 

2A  2A 

0,         —  cos  C,         —   cos  B. 


402,  403.]  TRILmEAR  SYSTEM.  309 

4.  The  co()rdinates  of  the  centre  of  tlie  circle  inscribed  in  the  tri- 
angle of  reference  are 

2J 

5.  The  equations  to  tlie  niedials,  (337,  2),  are 

aa  =  hfj,         })fi  —  cy,         cy=  aa. 

Since  any  one  of  these  is  the  consequence  of  the  other  two,  it  fol- 
lows that  the  medials  of  a  triangle  meet  in  a  point,  the  coordinates  of 
which  point  must  satisfy  the  three  equations  simultaneously. 

6.  The  equations  to  the  perpendiculars  of  a  triangle,  (340,  3),  are 
cos  A  —  p  cosi),         ft  cos  B—y  cos  C,         y  cos  0  —  oc  cos^. 

From  which  it  appears  that  the  altitudes  meet  in  that  point  which 
satisfies  the  three  equations  simultaneously. 

402.  Bisectors. — Every  point  on  the  line  bisecting  the 
angle  C,  gives  the  equation 

Hence  the  equations  of  the  internal  bisectors  are 

a^ji,         ft  =  y,  r=a; 

and  since  each  equation  is  the  consequence  of  the  other  two, 
it  follows  that  the  angle-bisectors  of  a  triangle  meet  in  a  point, 
which  point  is  the  centre  of  the  inscribed  circle,  the  coordi- 
nates for  which  must  satisfy  the  three  equations  simulta- 
neously. 

The  equations  for  the  external  bisectors  will  be 

«'  +  y5  =  0,         /3+  y=0,         a  +  y^O; 

and  hence  they  do  not  meet  in  a  point. 

The  external  bisectors  of  C  and  A,  and  the  internal  bisec- 
tor of  B  give 

a  +  ^  =^0,  /3  +  y^O,  a  -y  =  0, 

the  last  of  which  is  a  consequence  of  the  other  two ;  hence, 
the  external  bisector  of  two  angles  of  a  triangle,  and  the  internal 
bisector  of  the  other  angle  meet  in  a  point,  which  point  is  the 
centre  of  an  enscribed  circle. 

403.  Equation  of  a  Line.— Let  a^,  A,  y\,  be  the  tri- 
linear  coordinates  of  one  point  of  the  line,  «-;,  /?2,  ;K2,  the 
coiirdinates  of  another  point,  and  a,  ft,  y,  the  coordinates  of 
any  point.    Then  by  similar  triangles  (which  the  student  can 


310  MODERN  GEOMETRY.  [404. 

easily  construct),  we  find  the  relations  between  these  coordi- 
nates, and  the  resulting  equation  can  be  put  under  the  form 

la  +  mft  +  ny  =  0,  (1) 

in    which   l^/{^i,/3..,  y^,  y^),     m  =f  {a„  a.,  y„  y^),    and 

n=  f{ch,  ^y-:,  A,  A)- 

Equation  (1)  is  the  general  equation  of  a  right  line  in  the 
trilinear  system.  We  might  now  proceed  to  find  the  equa- 
tion of  a  line  passing  through  a  point  and  parallel  to  a  given 
line  ;  determine  the  condition  of  parallelism ;  determine  the 
distance  between  two  points  ;  the  equation  of  a  line  perpen- 
dicular to  a  given  one,  etc.;  but  these  must  be  left  as  exer- 
cises for  the  student. 

404.  Abridged  Notation. — The  equation  of  a  line  in 
terms  of  the  perpendicular  from  the  origin  is,  (30,  (6)), 

X  cos  a  +  y  sin  a  —  p  =  0. 

This  is  referred  to  as  '  the  line  cf,'  since  this  notation  at 
once  calls  to  mind  the  original  equation,  and  the  equation  to 
this  line  is  written 

a  =  0. 
Similarly,  /3  =  0 

is  the  equation  of  the  line  /?.  The  sides  of  the  triangle  of 
reference,  are  generally  referred  to  by  means  of  this  abridged 
notation;  thus 

a-  =  0,        /?  =  0,        y  =  0, 

being  the  equations  to  the  sides  of  the  triangle  of  reference, 
then 

la  +  mj3  +  ny  =  0, 

will  be  the  equation  to  a  right  line. 

As    an  application   of  this   method,   take  Example    5, 
M  L  page  263.      Let  ABC  be 

the  triangle  of  reference, 
then 

a  =  0,     /3  =  0,     y=0, 

will  be  the  equations  of 
its  sides. 

The  equations    of  the 
lines  will  be  of  the  form 


404.]  TRILINEAR  SYSTEM.  311 

for  B'C  la  +  m/3  +  ny  =  0,  (1) 

for  G  'A'  la  4-  m(3  +  ny  —  Q,  (2) 

for  AB'  la  +  m/3  +  n'y  =  0,  (3) 

in  which  each  equation  differs  from  each  of  the  other  two  by 
two  of  its  coefficients.  By  the  conditions  of  the  problem  these 
lines  are  to  intersect  on  lines  radiating  from  0  and  passing 
through  the  vertices  A,  B,  C.  At  the  intersection  C"  of  B'C 
and  AC,  y  will  have  a  constant  value  for  all  lines  passing 
through  C  ;  hence  eliminating  ;/ between  (1)  and  (2),  we 
have 

{I  -I)  a+  (m-  m')/3  =  0  ;  (CO) 

which  is  the  equation  of  some  line  through  C  It  is  also 
the  equation  of  a  line  through  C,  for  every  line  through  any 
angle  of  the  triangle  of  reference  may  be  reduced  to  the 
form 

a  +  Jc^  =  0; 

hence  (CG)  is  the  equation  of  the  line  GG'.  Similarly, 
eliminating  ^  between  (1)  and  (3),  we  have 

(l  -I)  a+  {n-n)y  =  0,  {BB') 

which  is  the  equation  of  a  line  BB' ;  and  eliminating  a  be- 
tween (2)  and  (3),  gives 

{m  —  m)  13  +  {n  —  n')y=0,  {AA') 

which  is  the  line  AA'.  Since  the  last  equation  is  a  conse- 
quence of  (GG')  and  {BB'),  they  will  have  a  common  point. 
Such  values  may  therefore  be  assigned  to  the  coefficients  l', 
m,  n,  etc.,  in  (1),  (2),  (3),  as  will  cause  them  to  be  the  equa- 
tions of  the  lines  AB',  B'C',  AG'. 

At  the  point  of  intersection  L  of  the  lines  B'C'  and  BG, 
the  perpendicular  a  will  be  zero.  (Here  it  is  the  perpendicu- 
lar that  is  zero,  and  not  the  equation  of  the  line.)     Then  (1) 

gives 

mft  -\-  ny  =  0. 

Similarly,  for  the  point  M,/3  =  0,  and  (2)  gives 

la+  ny  ~0 ; 

and  for  N,  y  =  0,  and  (3)  gives 

la  +  m/i  =  0. 


312  MODERN  GEOMETRY.  [405^07. 

Adding  these  gives 

la  +  mft  +  ny  =(), 

whicli  is  the  equation  of  a  line,  and  it  contains  the  three 
points  L,  M,  N;  hence  the  intersections  of  the  correspond- 
ing sides  of  the  triangles  meet  in  a  right  line. 

405.  Triangular  System. — Dividing  the  equation 

aa  +  h^  +  cy  =  2z^, 
by  2  J  gives 

aa        h^        cy  _  ^ 

2^  "^  2J    "^  2Z  ~ 

Let  X-—         v=^-         z=^^ 

and  the  preceding  equation  will  become 

X  +  y  -\-  z=l, 

and  since  x,  y,  z,  are  connected  with  a,  /3,  y,  by  known  rela- 
tions, the  last  equation  may  be  called  the  equation  of  a 
point  P.  It  will  be  observed  that  x  is  the  ratio  of  the  area 
of  the  triangle  aa,  to  that  of  the  triangle  of  reference.  This 
system  is  called  triangular. 

406.  The  Quadrilinear  System  consists  in  the  use  of 
four  lines  of  reference. 

407.  Equation  of  the  Second  Deg:ree. — Every  equa- 
tion of  the  second  degree  (being  made  homogeneous)  (401), 
may  be  written  in  the  form 

un^  +  v^^  +  wy^  +  %i  §y  +  %)'ya  +  2w  a^  =  0. 

Intersecting  this  by  the  straight  line 

la  +  TO/?  +  ny  =  0, 

and  we  find  that  there  may  be  two  real  points  of  intersec- 
tion, two  coincident  points,  or  two  imaginary  points,  which 
results  correspond  with  those  obtained  by  a  similar  opera- 
tion in  Cartesian  geometry  ;  hence  every  equation  of  the  second 
degree  in  trilinear  coordinates  represents  a  conic  section. 


408.]  OTHER  SYSTEMS.  3I3 

Olher  Systems. 

408.  Twenty-two  Systems.— The  Rev.  Thomas  Hill, 
D.D.,  LL.D.,  late  President  of  Harvard  College,  in  the  year 
1857,  gave  a  list  of  twentv-two  systems  of  cocirdinates,  as  fol- 
lows.* Let ./'  and  //  be  the  coia-dinates  in  the  Cartesian  sys- 
tem,?- and  cp  the  polar  coia-dinates,  ,9  the  length  of  the  curve, 
p  its  radius  of  curvature,  e  the  angle  between  the  radius 
vector  r  and  the  tangent,  r  the  angle  between  the  tangent 
and  an  assumed  axis,  and  r  the  angle  between  the  normal 
and  the  same  axis.     Then  we  have  t 

(1)  y  =/{x).  (2)  r  =f{cp).  (3)  p  =/(k).  (4)  r  =/  (.). 

(5)  p  =f{s).  (6)  a  =/(<7.).  (7)  r  =/(./.).  (8)  r  =:/{x). 

{9)x=/{cp).  {10)x=/{s).  (ll)r=/(x).  il2)p=/ix). 

{13)x=/{s).  (14)£=/(r).  (15)r=/(r).  (16) /.  =/ (r). 

(17)  r  =/(.).  (18)p--=/(^).  (19)^=/(.).  (20)r=/(O. 

{21)p=f{8).  (22)  e  =/(.). 

To  transform  a  curve  from  one  of  these  systems  to  another 
often  involves  the  Calculus;  their  discussion,  therefore,  is 
unsuited  to  this  work.  The  following  remarks  in  regard  to 
these  systems  is  an  abstract  of  those  made  by  Dr.  Hill,  to 
which  we  have  added  some  examples  by  way  of  illustration. 

(1).  y=z/(.r).  This  is  the  ordinary  system  of  bilinear 
coordinates,  and  the  leading  system  discussed  in  the  first  147 
pages  of  this  work. 

(2).  This  is  the  polar  system  of  coordinates. 

(3).  This  is  a  system  of  circular  coordinates,  much  used 
by  Professor  Peirce.     Dr.  Hill  shows  that  the  equation 

p  =  A  sin^K 

includes  a  great  variety  of  curves,  such   as  the   catenary, 
parabola,  cycloid,  etc.| 

*  Proceedings  of  the   American  Association  for  the  Advancement  of 
Science,  11th  meeting  (1857),  p.  43,  12th  meeting  (1858),  p.  1. 
t  Mathematical  Monthly,  1858-9,  p.  363. 
X  Gould's  Astro.  Jour.,  Vol.  II.,  p.  84 


314  MODERN  GEOMETRY.  [408. 

(4).  This  is  substantially  the  Intrinsic  equation  of  a  curve. 
The  intrinsic  equation  of  a  curve  is  defined  to  be  the  rela- 
tion between  the  length  of  arc  of  the  curve  and  the  change 
of  direction  of  the  tangent.  If  the  arc  begin  at  s  =  0,  and 
the  tangent  at  that  point  be  the  line  of  reference,  the  equa- 
tion will  be 

which  is  the  same  as  (4)  reversed. 

EXAMPLES. 

1.  The  intrinsic  equation  of  the  circle  is  s  =  Rt. 

2.  Of  the  involute  of  the  circle,  s  =  ^Rr^. 

3.  Of  the  catenary,  s  —  c  tan  r. 

4.  Of  the  cycloid,  s  =  4:R  sin  r, 

K    nt  4.x.  V.  1  1      1      1  +  sin  r        j9  sin  r 

5.  Of  the  parabola,  s  =  -  p  log  z : +  ~ ^-r- . 

tr  -I         2^°1—  sinr      1  —  sm'r 

(5).  "  An  interesting  point  in  the  5th  system  is  the  ease 
with  which  a  curve  expressed  in  it  may  be  produced  by  the 
deformation  or  metamorphosis  of  a  curve  in  rectangular  co- 
ordinates. The  curve,  for  example,  p—f(s)  may  be  pro- 
duced from  the  curve  y  —/  (x),  by  conceiving  the  onlinates 
of  the  latter  curve  to  be  fastened  at  right  angles  to  the  axis 
of  X,  and  the  axis  of  x  to  be  afterwards  curved  until  the  or- 
dinates  become  tangents  to  a  new  curve  thus  generated. 
The  ordinates  thus  become  the  radii  of  curvature  to  the 
axis,  and  the  axis  becomes  the  curve  s.  This  can  be  done 
rudely,  by  drawing  y  —f{x)  upon  a  card,  bending  it  on  the 
axis  of  X  to  a  right  angle,  cutting  it  on  the  ordinates  into 
strips  down  to  the  axis,  and  then  laying  these  strips  over 
each  other  so  as  to  make  the  points  of  the  curve  become 
points  of  tangency  to  the  involute  of  the  new  curve  p  =  f  (s). 
Thus  the  straight  line  y  -■  Ax  will  give  the  logarithmic 
spiral /3  =  yl  s ;  the  parabola  ?/^=  4Pa-,  will  give  the  involute 
of  a  circle  p^=  4:As ;  the  hyperbola  xy=A  produces  the  beau- 
tiful volute  or  double  finite  spiral  ps  —A;  the  parabola  y=Aa? 
produces  the  spiral  p  ~  As^ ;  the  ellipse  Ax^  4-  Bif  =  1, 
produces  the  epicycloids  Af?  +  Bs^  =  1,  or  hypocycloids 
A^  +  Bp^  =  1,  which  are  cvcloids  when  A  =  B." 


409.]  OTHER  SYSTEMS.  315 


The  equation  of  the  cycloid  will  be  p—  V  A-  —  s-,  from 
which  it  Avoiild  appear  that  it  has  only  one  real  branch,  an 
example  showing  that  what  appears  to  be  imaginary  in  one 
system  may  be  real  in  another. 

(6).  These,  in  many  cases,  may  readily  be  reduced  to 
r  =f  (qj)  and  investigated  in  this  form.  Thus  f  =  —  gj  is 
a  straight  line  ;  e  =  cp,  a  circle  ;  e  =  —  }^  cp,  a  parabola ; 
€  =  —  l(rp  —7t),  a  cardioid ;  e  =  —  2(p,  an  equilateral  hyper- 
bola. 

(7).  The  equation  r  =  a^p  is  equivalent  to  f  =:  (7  —  1)  q) ; 
hence  f  =  6,  or  r  =  fp  +  h,  is  the  equation  of  a  logarithmic 
spiral ;  r  =  ir/>  is  the  equation  of  a  jDarabola. 

(8).  This  may  be  changed  to  rectangular  cor*rdinates, 
thus  r  =  V-^r  +  y-  —  f  (x).  Changing  y  =  sec  -r,  y  =  tan  x, 
y  =  log  X,  into  r  =  sec  x,  r  —  tan  x,  r  =  log  x,  is  singular  and 
beautiful. 

(9).  This  system  may  be  changed  to  polar  coordinates, 
thus,  X  —  r  cos  (p  ■-f{<p). 

(10).  x  —  B  cot  f  and  x  =  A  Vcot  .,-,  are  two  equations 
of  an  equilateral  hyperbola. 

(11).  In  this  we  have  for  the  equation  of  the  parabola, 
tan  r  =  Ax,  and  for  the  circle  sin  r  =  Ax. 

(12).  This  may  be  constructed  approximately  to  a  scale. 
The  straight  line  y  =  Ax  may  be  metamorphosed  into  p  —  Ax. 

(13).  The  straight  line  will  be  ax  =  s,  the  cycloid  x=  as^. 

(16).  The  equation  p  =  .^r  includes  the  logarithmic  spi- 
rals ;  and  p  =  A/-^  is  the  equation  of  the  equilateral  hyperbola. 

(17).  r  =  as  includes  logarithmic  spirals. 

(18).  Form  of  curve  not  easily  detected. 

(19).  The  equation  of  the  circle  is  (p  =  as. 

(20).  The  equation  r  =  as  is  equivalent  to  r  = ^  qj,  or 

to  £  —  — ^j^  ^,  the  latter  of  which  is  the  6th  system.     The 
equation  of  the  circle  is 

sin  £  =  a  cos  (log  r  +  6)\/  —  1- 
409.  Another  system  investigated  by  Dr.  Hill  consists  in 


316  MODERN  GEOMETRY.  [410. 

using  as  variables,  tlie  perpendicular  from  the  origin  upon 
tlie  tangent  and  the  variable  angle  between  this  perpendicu- 
lar and  a  fixed  axis.* 

410.  Other  systems  may  be  found  by  introducing  as 
variables,  the  tangent,  subtangent,  normal,  subnormal,  per- 
pendicular from  the  origin  on  the  normal,  of  different  lines 
having  a  fixed  relation  between  them,  etc. 

*  Proceedings  of  the  American  Association  for  the  Advancement  of 
Science,  1873  and  1875. 


APPENDIX  I. 


The  system  of  Quaternions  grew  out  of  an  effort  to  geometrize  the 
imaginaries  of  Algebra.  Hamilton  sought  to  establish  a  system  which 
would,  at  the  outset,  give  a  clear  interpretation  to  the  square  roots  of  7iega- 
tives,  without  introducing  considerations  so  expressly  geometrical,  as  those 
which  involve  the  conception  of  an  angle.  This  idea  led  hina  to  consider 
Algebra  as  the  Science  of  Pure  Time,*  and  an  essay  containing  his  views 
upon  the  subject  was  published  in  1835.  f  From  this,  as  a  starting  point, 
lie  proceeded  by  a  system  of  logical  reasoning  to  make  a  new  system  of 
mathematics  ;  and  the  invention  of  this  system  was  made  when  he  estab- 
lished the  relation  i'^  =j'  —  k'  —  —  1  ; 
ij--k,jk  =  i,ki-j; 
ji  =  —  k,  kj  =  —  i,  ik  =  —j  ; 
which  he  did  in  an  article  first  published  in  1843.  :j:  Out  of  this  grew  his 
system  of  Quaternions.  §  In  presenting  the  subject  in  the  preceding  pages 
we  have  completely  ignored  the  abstract  reasoning  given  by  him,  and  have 
presented  it  entirely  in  its  geometrical  aspects.  | 


*  Hamilton's  Lectitres  on  Quaternions,  Preface,  p.  (2). 

f  Theory  of  Conjugate  Functions  or  Algebraic  Couples  ;  with  a  Pre- 
liminary Essay  upon  Algebra  as  the  Science  of  Pure  Time. — Trans,  of  the 
Royal  Irish  Academy,  Vol.  xvii..  Part  II.,  pp.  293-422. 

X  Proc.  Royal  Irish  Academy,  1848. 

§  Hamilton's  Lectures  on  Quaternions,  1853.  Also  the  following  Articles 
by  Hamilton  in  the  Philos.  Mag.  and  Jour,  of  Science,  (Lond.) : — Vol.  xxv., 
1844,  pp.  10,  241,  489  ;  Vol.  xxvi.,  1845.  p.  220;  Vol.  xix.,  pp.  26.  113,  326; 
Vol.  XXX.,  p.  458;  Vol.  xxi.,  pp.  214,  278,  511  ;  Vol.  xxxii.,  p.  367;  Vol. 
xxxiii.,  p.  58  (Brit.  Assoc.  Rep.,  1844,  Pt.  2,  p.  2) ;  Vol.  xxxiv.,  pp.  1^94,340, 
425  ;  Vol.  XXV.,  pp.  133,  200;  Vol.  xxxvi.,  p.  305  ;  Vol.  iii.,  1852,  p.  371 ; 
Vol.  iv.,  1852,  p.  303  ;  Vol.  v.,  1853,  pp.  117,  236,  321  ;  Vol.  vii.,  1854,  p. 
492  ;  Vol.  viii.,  1854,  pp.  125,  261  ;  Vol.  ix.,  1855,  pp.  46.  280.  Vols,  iii.,  iv., 
v.,  are  on  continued  fractions  in  Quaternions.  Vols,  vii.,  viii.,  ix.,  on  some 
extensions  of  Quaternions.  See  also  Nichol's  Cyclopcedia  of  Physical  Science, 
article  Quaternions,  which  article  was  prepared  by  Hamilton  and  is  written 
in  a  remarkably  free  and  easy  style,  pp.  706-726. 

I  Hamilton's  Lectures,  p.  72. 

317 


318  APPENDIX. 

Tliere  were  many  attempts  during  the  half  century  preceding  Hamilton's 
work,  to  accomplish  the  same  result.  In  order  to  understand  more  clearly 
the  nature  of  the  problem,  it  is  necessary  to  go  back  and  consider  what  had 
been  accomplished  by  algebraic  analysis. 

The  corner-stone  of  algebra,  or,  at  least,  that  which  made  it  a  distinc- 
tive science,  was  the  establishment  of  the  law  of  minus.*  According  to 
algebra,  we  not  only  understand  the  meaning  of  the  symbol  minus  when  it 
indicates  an  operation,  but  we  also  know  how  to  interpret  it  when  it  appears 
in  the  result  of  the  solution  of  a  problem.  The  law  thus  determined  was 
universal,  being  applicable  to  all  kinds  of  problems,  geometrical,  physical, 
imd  abstract.     In  the  course  of  analysis,  a  new  expression  appeared,  the 

V  —  1 ,  which,  in  an  algebraic  sense,  is  an  expression  for  an  impossible  opera- 
tion.   It  is  impossible  to  find  a  number  whose  square  is  negative,  and  the 

V  —  a  (a  being  always  positive)  is  an  expression,  the  operation  indicated  by 
which  cannot  be  performed.  Still,  there  are  many  expressions  resulting 
from  the  application  of  algebraic  analysis  to  geometrical  problems  contain- 
ing the  imaginary,  the  values  of  which  are  known  to  be  real  ;  as,  for  in- 
stance, Euler's  formulas,  which  are 

2v— Isina;  —  e  —  e  , 

2  cos  a;  =  e^^ -^  +  6~**^~ ^ ; f 
and  De  Moivre's  formula,  which  is 


(cos  X  +  V  —  1  sin  a;)™  =  cos  mx  +  V  —  1  sin  mx  ; 

all  of  which  appear  to  be  imaginary,  but  are  actually  real  for  all  values  of 
X,  in  which  x  is  expressed  as  an  arc  of  a  circle,  the  radius  being  unity. 
These  formulas  (and  many  others)  are  the  results  of  pure  analysis,  and  it  is 
very  natural  to  seek  for  a  geometrical  interpretation  of  them.  Hamilton's 
Quaternions  furnishes  an  easy  and  natural  mode  of  interpreting  the  V  —  1 

*  Algebra  has  finally  come  to  be  '  The  Science  of  the  Equation.' 
f  Let  X  V  —  1   —X,  and  the  equations  become 

2  V^nr  sin  X'  l/^T  =  6^'  -  e~*' 
2  cos  X  V"^^  =  e^'  +  e~*', 
the  second  members  of  which  are  in  the  form  of  real  quantities.     Some 
writers  put  Sin    x  for    V  —  1  sin  a;  -»/  —  1,  and  Cos  x  for  cos  x  V  —  1,   and 
call  the  expressions  Sin  x  and  Cos  x  potential  functions.     Dropping  the 
accents  from  the  preceding  expressions,  we  have 

2  Sm  x  =  e    —  e     , 
2  Cos  a;  =  e^  +  e~'^, 
and  Cos  x  +  Sin  x  =:  e*. 


APPEyDTX.  319 

when  it  is  the  result  of  an  operation  in  that  system,  but  it  does  not  cover 
the  imaginaries  of  ordinary  algebra  (Art.  355).  The  interpretations  of  nega- 
tives and  of  imaginaries  differ  widely  in  this  regard  ;  for  while  the  former 
covers  all  cases  which  have  been  known  to  arise,  tlie  latter,  as  explained  by 
(Quaternions,  is  applicable  only  to  the  system  which  was  invented  expressly 
for  the  purpose  of  giving  them  a  rational  existence.  A  nionieut's  considera- 
tion, however,  will  show  that  a  different  result  could  hardly  be  expected  ; 
for  it  is  found  that  tlie  negatives  in  algebra  always  have,  in  a  certain  sense, 
a  real  existence,  while  the  imaginaries  have  not.  Thus,  in  algebra,  if  a 
negative  result  does  not  agree  witli  the  wording  of  the  problem,  we  know 
that  by  stating  the  problem  in  an  opposite  sense  the  result  becomes  real ; 
but  when  an  imaginary  occurs  in  the  solution  of  an  algebraic  problem,  we 
cannot,  by  any  transformation  of  the  language  make  the  result  real ;  it  is 
necessary  to  change  the  data.  To  illustrate  by  means  of  some  examples  : 
take  first,  the  following, 

A  cistern  can  be  filled  in  56  minutes  by  two  faucets  flowing  togciher ;  if  they  flow 
separately,  it  will  take  one  faucet  66  minutea  longer  to  fill  the  cistern  than  the  other:  in  what 
time  will  the  cistern  be  filled  by  each  ? 

Let  X  be  the  time  required  to  fill  it  from  the  larger  one,  then  will  x  -i-  66 
be  the  time  required  by  the  other,  and  we  have  the  equation 

56^        J6     ^  J 
X        a;  +  66  ~    ' 

by  solving  which  we  find  a*  =  88  or  —  42,  and  .r  -f-  66  =  154  or  24  ;  hence  the 
required  times  are  88  and  154  minutes  respectively;  and  —  42  and  24  minutes 
respectively.  The  last  results  are  incompatible  with  the  statement  of  the 
problem  ;  but  if  we  consider  that  —  42  is  the  time  required  for  the  first  to 
empty  the  cistern  if  it  were  full  and  permitted  to  flov  out  through  that 
faucet,  and  24  the  time  required  for  the  other  to  fill  it  ;  the  results  become 
real.  By  using  these  numbers,  it  will  be  found  that  if  both  faucets  be 
opened  at  the  same  time  the  cistern  being  empty  and  is  being  filled  by  the 
latter  while,  at  the  same  time,  it  flows  out  through  the  former,  it  will  re- 
quire 56  minutes  to  fill  it. 

Next,  take  the  following  example, 

The  sum  of  the  times  required  for  two  faucets  to  fill  the  cistern  is  8  minutes,  and  the 
product  of  the  times  is  20  minutes  ;  required  the  times. 

Here  we  have  the  equations 

jr  +  y  =    8, 

xy  -  20, 

a  solution  of  which  gives  a;  =  4  ±2  V^  —  1  and  y  =  4T2'/— 1  ;  hence  the 
conditions  are  impossible,  and  no  interpretation  of  the  language  will  make  it 
real.  By  changing  the  data  by  making  the  sum  of  the  times  9,  or  the  pro- 
duct 15,  the  problem  becomes  real. 

The  neQotivea  and  imaginaries  of  algebra  are  not  considered  as  quanti 


320  APPENDIX. 

ties  ;  *  they  are,  primarily,  symbols  of  unexecuted  operations,  but  when  they 
appear  in  the  results  of  analysis,  their  chief  use  is  to  aid  in  the  interpreta- 
tion of  the  problem  from  which  they  were  deduced.  In  regard  to  the 
attempts  to  geometrize  the  imaginaries  and  the  views  held  in  regard  to  them 
we  make  a  few  extracts  from  Hamilton's  writings,  and  from  the  references 
therein  given. 

Dr.  Wallis  of  Oxford,  in  his  "  Treatise  of  Algebra,"  published  in 
1685,  proposed  to  interpret  the  imaginary  roots  of  a  quadratic  equation,  by 
going  out  of  the  line,  on  which  if  real  they  should  be  measured.  He  says 
"so  that  whereas  in  the  case  of  Negative  roots,  we  are  to  say  the  point  B 
cannot  be  found,  so  as  is  supposed  in  AC  forward,  but  backward  it  may  be 
in  the  same  line  ;  we  must  here  say,  in  the  case  of  a  Negative  Square  the 
point  B  cannot  be  found  so  as  was,  in  the  line  AC;  but  above  that  Line  it 
may  be  in  the  same  Plane.  This  I  have  the  more  largely  insisted  on,  be- 
cause the  notion  (I  think)  is  new  ;  and  this,  the  plainest  Declaration  that  at 
present  I  can  think  of,  to  explicate  what  we  commonly  call  the  Imaginary 
Roots  of  Quadratick  Equations.     For  such  are  these." 

In  June,  1805,  M.  Abbe  Buee  read  a  paper  entitled  Memoircs  sur  lea 
Quantites  Imaginaires,  which  was  printed  in  the  Philosophical  Transactions 
(London)  for  1806.  This  writer  has  the  credit  of  being  the  first  to  formally 
maintain  that  the  V  —  1  as  a  symbol  denoted  perpendicularity,  though 
this  view  had  been  suggested  by  others.  This  view  has  never  led  to  the  de- 
velopment of  a  system,  and  in  explaining  his  methods,  Buee  expressly 
excludes  the  consideration  of  the  position  of  the  factor-lines  in  multiplica- 
tion. 

The  preceding  are  simply  interpretations  of  the  imaginary.  Hamilton 
gives  full  credit  to  M.  Argand  as  being  the  first  writer  to  multiply  together 
(as  well  as  add)  directed  lines  in  one  plane  ;  which  he  did  in  an  "  Essay  on  a 
manner  of  representing  Imaginary  Quantities,"  published  in  1806.  This 
method  was  reproduced  independently  by  a  Mr.  Warren  in  1828,  and  in  the 
same  year  by  M.  Mourey  in  a  work  entitled  :  "  The  True  Theory  of  Nega- 
tive Quantities  and  of  the  so-called  Imaginary  Quantities  "  (Paris,  1828).  As 
the  method  of  Argand  is  of  considerable  interest  historically,  we  here  illus- 
trate its  method  by  the  following  extract  taken  from  Mourey 's  work. 

Take  the  expression  a  +  &  V~\,  and  give  it,  at  first,  the  form 


Va^  +  ¥ 


■+  1-1 


_  Va-  -ib'        \'a'-  4-  6- 


If  we  take  the  sum  of  the  squares  of  the  fractions,  which  are  between  the 
brackets,  we  find  that  this  sum  is  equal  to  1  ;  and  from  thence  we  conclude 
that  these  two  fractions  can  be  regarded  as  being  the  sine  and  cosine  of  the 
same  angle  a.  Designate  also  the  modulus  Va«  -|-  6*  by  A  ;  the  imaginary 
expression  can  be  put  under  the  form  A(cos  a  +  V—\  sin  a).  Considering 
that  this  expression  contains  really  but  two  quantities,  the  modulus  A  and 

^  Lectures,  p.  (2). 


APPENDIX.  321 

the  angle  a,  M.  Mourey  proposed  to  regard  the  modulus  A  as  expressing  the 
length  of  a  right  line  OA,  and  a  as  being  the  angle 
AOB,  which  this  line  malves  with  a  fixed  axis  OB. 
In  other  words,  the  modulus  A  represents  a  line 
of  a  certain  length,  which  at  first  lay  upon  the  axis 
OB,  and  which,  by  maldng  a  movement  round  the  Fig.  277 

origin  0  upward,  has  departed  from  this  axis  by  an  angle  a.  M.  Mourey 
gives  the  name  terser  to  this  angle,  or,  rather,  to  the  arc  which  measures 
it ;  and  then,  instead  of  the  imaginary  expression,  he  writes  simply  Aa,  a 
notation  very  suitable  to  recall  at  the  same  time  the  modulus  A  and  the  'oer- 
ser  a.  He  proposes  even  to  give  the  name  route,  or  way,  to  the  length 
OA,  placed  in  its  true  position  with  regard  to  OB,  so  that  A  verser  a,  or 
Aa,  is  the  route  from  0  toward  A. 

As  a  line  can  make  around  the  origin  0  as  many  revolutions  as  we 
please,  and  that,  also,  as  well  by  commencing  its  rotation  below  as  well  as 
above  OB,  it  follows  that  the  verser  may  pass  through  all  states  of  magni- 
tude, and  be  as  well  negative  as  positive.  It  will  be  positive  when  the 
movement  of  the  line  shall  have  commenced  above  ;  it  will  be  negative 
when  the  movement  commenced  below.  From  this  it  follows  that  the  same 
route  can  be  represented  with  a  verser  which  is  positive,  or  one  which  is 
negative,  provided  that  the  sum  of  the  versers,  abstraction  being  made  of 
the  signs,  is  3G0\ 

From  the  preceding  conventions  it  results  that  a  way  can  be  represented 
by  giving  to  the  length  A  an  infinity  of  diflFerent  versers.  Suppose,  to  fix 
the  ideas,  that  OA  should  be  a  determinate  way,  and  that  then  the  verser 
AOB,  should  be  an  acute  angle  or ;  it  is  evident  that  the  position  of  OA 
will  undergo  no  change  if  we  add  or  subtract  from  a  any  number  whatever 
of  entire  circumferences.  Thus  is  established  this  important  remark,  that 
if  we  designate  by  27t  an  entire  circumference,  or  360",  and  by  n  any  whole 
number  whatever,  positive  or  negative,  the  expression  A{27tn  +  a)  will 
represent  the  same  route  as  Aa  ;  this  is  expressed  by  the  equality 

A(27fn  +  a)  =  Aa. 

When  we  give  to  ^  a  verser  equal  to  zero,  the  length  A  lies  upon  the 
line  OB.  When  the  verser  is'  equal  to  tt  or  180°,  this  length  is  found  in 
the  opposite  direction,  OX ;  then  it  is  nothing  else  than  the  negative 
quantity  —  A.  Thus  we  ought  to  regard  as  altogether  equivalent  the  two 
expressions  —  A  and  Ajt. 

After  these  preliminaries,  M.  Mourey  establishes  the  rules  of  algebraic 
calculus.  Next  determine  the  rule  to  be  followed  in  the  multiplication  of 
any  two  quantities  whatever,  Aa  and  Bf:i.  Here  the 
two  factors  are  the  magnitudes  A  and  B,  measured 
upon  two  lines  OA  and  OB,  which  make,  with  a  fixed 
axis  OX,  angles  AOX,  BOX,  represented  by  the  ver- 
sers a  and  /3.  (The  reader  can  draw  any  line  through 
0  to  represent  OX.)  It  is  necessary,  then,  first  of  all, 
to  give  to  the  definition  of  multiplication  the  exten- 
sion suitable  to  render  it  applicable  to  the  case  in  ques-  ^^*''  ^^ 


322  APPENDIX. 

tion.  But,  considering  that  tlie  multiplier  B/5  indicates  a  line  B,  whicli 
departs  from  the  fixed  line  OX  hj  an  angle  equal  to  fi,  M.  Mourey  regards 
multiplication  as  having  for  its  object  to  take  at  first  the  length  A  in  its 
actual  direction  as  many  times  as  there  are  units  in  B,  giving  the  line  OA' , 
and  to  turn  the  new  line  OA'  around  the  point  0,  to  depart  from  this  direc- 
tion by  an  angle  equal  to  fi,  and  to  give  it  the  position  OC.  From  this  it 
follows  that,  in  designating  by  AB  the  product  of  the  two  magnitudes, 
abstraction  being  made  of  all  idea  of  position,  the  product  sought  will  be 
AB{a+(S).    Thus  we  have 

Aa-xiBfi  =  AB{a  +  P); 

that  is  to  say,  we  multiply  the  moduli  according  to  the  ordinary  rules  of 
arithmetic,  and  take  the  sum  of  tJie  versers. 

If  the  versers  are  each  equal  to  Tt  or  180°,  we  shall  have  J.;r  x  B7C=AB{27t). 
But  A7t  and  Btt  are  nothing  else  than  —  A  and  —B,  and  ABlrt  is  the 
same  thing  as  +  AB ;  then  —A  x  ~B=i+AB.  This  is  the  known  rule, 
—  ^y  —  gives  + , 

According  to  this  rule,  the  square  of  Aa  will  be  A^{2a) ;  that  is  to  say, 
we  take  th£  square  of  the  modulus  and  double  the  verser.  Then,  reciprocally, 
the  square  root  is  obtained  by  extracting  tJie  square  root  of  the  modulus  with- 
out regarding  the  verser  ;  then  take  half  the  terser. 

Let  us  come  now  to  the  interpretation  of  the  imaginary  expression 
^ —A^.  For  this  purpose,  let  us  observe,  first,  that  it  is  equivalent  to 
Vj.^(2n.7r-t-  jr) ;  then  extracting  the  square  root. 


'^-A^=A{mt  +  \Tt\ 

If  n  is  even,  the  verser  n%  +  \Tt  places  the  length 
A  in  the  same  position  as  \Tt ;  that  is  to  say,  in  the 
position  0  Y,  perpendicular  to  OX.  If  n  is  uneven, 
the  verser  mc  +  lit  will  place  the  length  ^  in  a  posi- 
tion OT' ,  perpendicular  to  OX,  but  below.  Thus, 
in  the  system  of  M.  Mourey,  the  expression  \f —A'^ 
offers  no  longer  to  the  mind  any  idea  of  impossibility. 
It  represents  two  routes,  OY  and  OY' ,  equal  and 
opposite,  both  perpendicular  to  the  fixed  axis  OX. 
We  see  shadowed  here  some  of  the  elements  of  Hamilton's  Quaternions, 
still  the  systems  have  very  little  in  common.  The  difference  between  them 
is  great.  In  Mourey's  system  all  the  operations  are  in  one  plane,  two  dimen- 
sions only  of  space  being  necessary  ;  and  the  multiplication  of  two  lines 
produces  a  third  in  the  same  plane.  In  this  system  it  is  necessary  to  know 
the  angle  between  the  lines  and  another  fixed  line,  and  if  the  axis  coincides 
with  one  of  the  lines,  the  multiplication  of  the  two  lines  will  be  expressed 
by  laying  off  on  one  of  them  the  numerical  product  of  the  two.  It  is  quite 
evident  that  Hamilton  recei\  ed  little  or  no  aid  from  these  writings  in  the 
establishment  of  his  system ;  though  some  may  have  served  as  hints,  but 


APPEXDIX.  323 

nothing  more.  Most  of  tlie  other  important  investigations  which  have  a 
bearing  upon  the  subject,  were  contemporary  with  Hamilton's  work.  It  is 
proper  to  note  that  M.  Servois  seems  to  have  made  the  nearest  approach  to 
an  anticipation  of  quaternions.*  He  inferred  from  analogy,  that  if  ex,  (i,  y, 
be  the  angles  between  a  right  line  and  the  three  rectangular  axes,  the 
following  expression  ought  to  be  true  : 

(^costt+^-cos/J+rcosjK)  {p'  cosa+(?'co3  j'J  +  r'cos^)  = 

cos^  a  +  COS'*  /?+  cos-  y  —\\ 

but  he  could  not  determine  the  values  of  p,  q,  r,  p' ,  q  ,  r ,  and  asked 
"  Will  they  be  imaginaries,  reducible  to  the  general  form  A  +  B  V^—i.  ?  " 
It  is  now  known  that  they  are  identical  with  the  +i,  +_;,  +k,  —i,  —j,  —k, 
of  quaternions. 

M.  Cauchy,  in  his  Cours  d' Analyse  (Paris,  18'21),  remarks,  "Every 
imaginary  equation  is  only  the  symbolic  representation  of  two  equations 
between  real  quantities." 

We  would  do  great  injustice  to  Hamilton's  worthy  friend  Mr.  John 
T.  Graves,  were  we  to  make  no  mention  of  his  labors.  He  seemed,  if 
possible,  to  be  the  more  enthusiastic  of  the  two,  in  trying  to  overcome  the 
difficulties  which  beset  these  men  in  their  endeavors  to  construct  the  new 
system.  If  his  labors  were  not  crowned  with  being  the  successful  inventor, 
he,  at  least,  encouraged  Hamilton  in  his  labors,  f  and  brought  to  his  notice 
the  results  of  his  predecessors  and  of  his  contemporaries.  Still,  however 
much  he  may  have  profited  by  his  suggestions,  or  by  the  suggestions  of 
others,  Hamilton  appears  justly  to  have  the  full  credit  of  conceiving  and 
first  applying  the  fundamental  principles  of  his  system. 

We  add  a  few  words  more  upon  the  invention.  Hamilton  supposed 
that  he  could  retain  the  commutative  principle,  \  or  the  interchangeability 
of  the  factors  in  regard  to  order  ;  but  after  many  trials,  with  expressions 
of  various  forms,  he  was  obliged  to  abandon  the  principle,  and  make 
ij  ==  ~  ji.  He  had  previously  assumed  that  i-  =  j'*  =  —  1  ;  but  the  value 
of  k,  which  appeared  later  in  the  discussion,  had  not  been  fixed.  At  first 
he  tried  A*  =  4- 1,  because  i^j^  ~  -\-\.  but  as  this  failed  to  work,  he  tried 
A;^  =  —  1,  and  thus  completed  his  fundamental  assumptions^  for  the  mul- 
tiplication of  two  vectors.  Vectors  generally  are  represented  by  the  letters 
of  the  Greek  alphabet,  but  I  presume  that  Hamilton  used  the  Roman  and 
italic  letters  for  the  mutually  perpendicular  vectors,  because  the  letter  i 
had  long  been  used  in  analysis  for  the  V—1,  and  also  because  his  friend, 
Mr.  Graves,  was  using  it,  and  had  in  some  of  his  investigations  made 
i^z=  -1,  andp  =  _l.| 

*  Lectures,  p.  (57).  f  Lectures,  Preface,  p.  (35). 

:j:  Lectures,  Preface,  p.  (43).  §  Lectures,  Preface,  p.  (46). 

5  Hamilton  was  not  the  only  one  to  invent  a  non-commutative  system. 
Professor  H.  Grassman,  in  1844,  the  year  following  that  of  the  first  publi- 
cation by  Hamilton  of  his  Quaternions,  published  a  very  original  and  re"- 
markable  work  (Aus  Dehnungslehre,  or  Doctrine  of  Extension),  involving  the 


324  APPENDIX. 

The  development  of  this  subject  may  be  considered  in  another  liglit. 
By  examining  the  successive  processes  in  mathematics  we  find  that  it  lias 
been  extended  by  the  removal  of  restrictions.  This  principle  makes  its 
appearance  in  the  earliest  stages  of  mathematics,  and  may  be  recognized  in 
all  departments  of  the  subject ;  and  may  be  applied  to  many  of  the  princi- 
ples of  quaternions.*  We  will  illustrate  the  principle  by  means  of  a  few 
examples. 

In  arithmetic  we  are  taught  that  multiplication  is  a  short  process  of 
making  repeated  additions.  According  to  this  definition  multiplication  can 
be  performed  by  a  process  of  counting.  It  answers  well  so  long  as  the 
numbers  are  positive  integers,  but  fails  when  the  multiplier  is  a  proper 
fraction.  In  order,  therefore,  to  multiply  by  a  fraction  it  is  necessary  to 
make  a  new  definition,  or  extend  the  meaning  of  the  old  one ;  and  the  latter 
alternative  is  the  one  chosen.  In  the  extended  sense,  multiplication  is  the, 
process  of  finding  the  result  of  taking  one  number  several  times,  or  part  or 
parts  of  a  time.  The  pi'ocess  of  involution  is  a  further  extension  of  the 
principle,  being  a  process  of  repeated  multiplication,  and  hence  is  the  repeat- 
ing of  the  result  of  the  result  of  repeated  additions.  When  logarithms 
are  employed,  the  original  conception  of  addition  is  almost  entirely  lost  sight 
of ;  indeed  to  a  certain  extent  the  process  is  reversed,  multiplication  being 
performed  by  addition  hj  the  aid  of  logarithms.  When  the  exponent  is 
literal,  it  is  read  as  a  power,  thus  10^  is  read  '  the  x  power  of  10,'  but  if  the 

exponent  is  fractional  its  meaning  is  entirely  changed.     If  a;  =  §,  we  have 

a 
10 ',  which  arithmetically  is  the  square  of  the  cube  root  of  10,  but  by  the 

removal  of  restrictions  the  definition  is  extended  so  that  a  potcer  is  defined  to 

be  a  number,  entire  or  fractional,  positive  or  negative,  wJiich  is  placed  at  the 

right  and  above  another  number.     According  to  this  definition  we  read  the 

expression  'ten  to  the  two-thirds  power.'    This  reading  indicates  t]xeform 

of  the  expression  but  not  the  arithmetical  operations  to  be  performed  in 

—  X  — ^ 

reducing  it.  The  same  remark  applies  to  the  reciprocals,  as  a  ,10  ^, 
etc. 

In  algebra,  negative  qualities  are  considered,  and  multiplication  includes 
the  process  of  performing  repeated  subtractions ;  but  instead  of  making 

multiplication  of  inclined  lines  (aussere,  outer,  multiplikation),  which  had 
the  Tion-commutatim  principle.  Neither  Grassman  nor  Hamilton  owed  any- 
thing, in  their  original  papers,  to  the  works  of  the  other,  and  the  methods  of 
the  two  are  quite  distinct  from  each  other.  The  work  of  Grassman  has  been 
admired  by  mathematicians  since  his  day.  Hamilton  speaks  of  it  in  the 
highest  terms  as  an  original  work,  and  Professor  Clifford,  in  an  article  enti- 
tled, "Applications  of  Grassman's  Extensive  Algebra,"  in  the  American 
Journal  of  Mathematics,  Vol.  I.,  No.  4,  p.  350,  endeavors  to  determine  the 
place  of  Quaternions  in  the  more  extended  system  ;  and  a  demonstration  that 
the  algebra  obtained  by  a  generalization  of  the  laws  of  more  extended  sys- 
tems of  algebras  is  always  a  compound  of  quaternion  algebras  which  do  not 
interfere  with  one  another. 

*  Hamilton's  Lectures,  Preface,  p.  (50). 


APPENDIX.  •  325 

a  new  definition  to  cover  tliis  idea,  the  old  de6nition  is  extended  by  remov- 
ing the  restrictions  which  were  previously  given  to  it,  and  with  the  new  idea 
it  covers  the  case  of  minus  by  minus,  or  of  subtracting  the  result  of  repeated 
subtractions.     If  the  exponent  is  zero,  we  have 

a  =  1, 
whatever  be  the  value  of  a,  and  if  a  also  is  zero  we  have  the  expression 

which,  according  to  the  arithmetical  definition  of  number,  has  no  meaning. 
Yet  in  the  Differential  Calculus  we  not  only  meet  Avith  this  expression,  but 
also  such  as 

1      ,      U     X       00,       -ly.,         CD    , 

and  they  are  interpreted  as  having  rational  meanir-gs.  Arithmetically  quan- 
tity is  something  which  can  be  measured,  and  number  is  employed  to  express 
the  measure.  Originally  it  implied  'how  many,'  as  5,  9,  13,  etc.,  but  by  an 
extension  it  includes  fractions  ;  it  includes  all  measurements  which  can  be 
expressed  by  Tneans  of  ^(/wres.  But  0  expresses,  arithmetically,  the  absence 
of  all  quantity,  and  hence,  technically,  would  have  no  place  in  the  science 
of  mathematics  were  it  not  for  the  principle  of  extension.  For  the  same 
reason  infinites  would  find  no  place ;  but  as  defined,  both  form  exceed- 
ingly important  elements  in  this  science.  Zero  (so  called)  in  the  Calculus 
is  not  an  arithmetical  zero  ;  it  is  an  infinitesimal.  Shall  Ave  form  a  ncAV 
symbol  for  the  new  meaning,  or  shall  we  extend  the  definition  so  as  to 
give  a  new  meaning  to  the  old  symbol  ?  The  former  might  have  been  done, 
bat  the  latter  was  chosen.  The  restrictions  which  had  been  given  to  it 
were  removed,  and  the  definition  thus  extended.  Similarly,  the  infinites, 
according  to  the  original  conception  of  number,  do  not  exist  ;  but,  as 
defined,  they  are  realities.  According  to  this  view,  we  might  found  the 
Differential  Calculus  (substantially  as  many  writers  have  done)  after  this 
manner  : — In  arithmetic  we  have  considered  only  one  class  of  numbers, 
cailed  finite  and  discontinuous,  as  3,  12.  f,,  etc.,  Avith  Avhich  our  daily  expe- 
rience makes  us  familiar.  But  we  find  that  great  power  is  given  to  mathe- 
matics by  including  other  numbers  which  we  Avill  noAv  proceed  to  define  and 
to  which  we  wUI  assign  certain  laws.  One  of  these  we  Avill  call  Infinitesi- 
mal. This  we  define  to  be  so  small  (and  if  you  please,  of  such  a  peculiar 
character),  that  it  cannot  be  added  to  an  arithmetical  number  ;  so  that  if  it 
be  represented  by  5  we  Avill  have 

5  +  S  =  5,7-S-7,a  +  8=:a,  etc. 

But  it  can  be  multiplied  by  an  arithmetical  number,  thus 

5x8=z5S,-7xS  =  -7d,axS=  aS,  etc. 

K  divided  by  an  arithmetical  number  the  quotient  will  be  called  zero: 
thus, 

S  fi 

5  =0,  -  =0.etc 


326  APPENDIX. 

but  if  an  arithmetical  number  be  divided  by  an  infinitesimal  tlie  result  will 
be  an  infinite,  tlie  laws  for  which  will  now  be  assigned.  Briefly,  we  would 
define  an  infinite  as  one  to  which  neither  a  finite  nor  an  infinitesimal  can 
be  added,  which  may  be  multiplied  by  a  finite,  which  cannot  be  divided  by 
&  finite  (or  if  so  divided  will  give  an  infinite  of  a  higher  order),  and  if  di- 
vided into  a.  finite  will  give  a  result  called  zero.  In  a  similar  manner  we 
would  define  infinites  and  infinitesimals  of  different  orders,  the  relations 
between  the  successive  orders  being  similar  to  those  just  defined.  Next  we 
would  proceed  to  determine  how  the  ratios  of  the  infinitesimals  of  the  first 
order  could  be  determined  from  given  relations  between  the  finites ;  the 
result  would  be  the  first  differential  coefficient. 

This  process  will  appear  (as  it  really  is)  arbitrary  to  the  student,  and,  at 
first,  would  doubtless  be  considered  by  many,  as  unworthy  of  being  called 
a  system  ;  still  as  an  application  of  the  principle  of  extension  by  the  removal 
of  restriction,  it  is  worthy  of  being  presented  to  every  student  while  studying 
the  calculus.  The  main  object  is  to  arrive  at  trutli,  and  hence  we  should 
not  be  confined  to  one  system,  or  mode  of  presenting  it.  In  the  light  of 
what  has  been  said  we  will  interpret  one  of  the  preceding  expressions  ;  thus 


means  (in  the  light  of  the  Calculus)  that  1  is  not  an  arithmetical  1  but  a 
thing  which  differs  from  1  by  an  infinitesimal,  and  may  be 

1  +  5, 

the  infinite  power  of  which  will  be  some  finite.  The  other  expressions  may 
be  interpreted  in  a  similar  manner.  Those  who  have  not  studied  Vanishing 
Fractions  may  not  understand  the  conclusions  here  given. 

But  to  proceed.  In  finite  geometry  the  parameters  are  considered  as  fixed 
quantities,  but  in  the  Calculus  these  are  sometimes  made  to  vary  ;  so  that  by 
the  removal  of  restrictions  we  state  that  a  geometrical  constant  is  one  which 
varies  infinitely  slow  compared  with  the  rate  of  change  of  the  variables.  Re- 
turning to  multiplication,  we  observe  that  multiplication  of  a  line  by  a  num- 
ber, as  3&,  is  represented  by  a  line  whose  length  is  three  times  as  long  as  h  ; 
but  we  are  not  restricted  to  tlii.s  representation,  for  the  product  of  a  into  &,  or 
ah,  is  represented  by  a  rectangle  ;  neither  are  we  restricted  to  these,  for  in  qua- 
ternions ij  is  represented  by  rotation.  We  observe  that  the  removal  of  restric- 
tions sometimes  imposes  new  restrictions.  Thus,  in  quaternions  restrictions 
were  removed  and  extensions  made  in  regard  to  the  algebraic  symbols  + ,  — , 
=,  X,  -J-,  but  they  resulted  in  destroying  the  commutative  principle  of 
algebra. 

But  we  must  bring  these  remarks  to  a  close,  as  they  have  been  already 
extended  far  beyond  what  was  originally  intended.  We  did  not  intend  to 
write  a  critical  history  of  the  subject,  but  simply  to  give  an  outline  of  the 
manner  in  which  the  subject  was  developed.  As  for  the  extent  of  the  use- 
fulness of  quaternions  we  express  no  opinion  ;  but  we  can  safely  assert  that 
no  principle  so  original,  novel,  and  comprehensive  as  that  given  by  Hamil- 
ton  can  be  introduced  into  any  science  without  yielding  much  good  fruit. 


APPENDIX.  327 

It  seems  that  Euler's  trigonometrical  formulas  ought  readily  to  be  de- 
duced from  Hamilton's  system ,  and  such  is  the  case  ;  but  we  could  not  in- 
troduce it  into  the  body  of  the  work  without  assuming  the  development  of 
log  (1  -+-  X),  and  the  value  of  ;r  as  given  by  a  certain  series.  But  now, 
assuming  the  development,  we  proceed  as  follows  : 

log  {\  +  x)  =  M\x  —  \x'-  +  \x^  —  \x^  +  etc.] 
log  {l-x)  =  M\_-x-^x^  -  \x^  -  \x^  -  etc.], 

the  latter  being  deduced  from  the  former  by  changing  xXo  —  x.     Subtract- 
ing we  have 

log  (1  -fx)  —  log(,l  -  a-)  =  2Mx  [1  +  ix^  +  W  +  etc.]. 


log 


l-\-x 


-  2Mx  [1  +  \x^  +  \x''  +  etc.]. 


Now  let  X  =  V—  1,  and  we  have 

(1   +     V^\)  H-  (1  -     \'^\)  =    \^^1, 

and  substituting  the  same  value  of  x  in  the  second  member,  we  have 

log  V^l  =  2M  1=1  [1  -  i  +  i  -  I  +  etc.]. 
The  quantity  within  the  brackets  is  \z*  therefore 

log  \^^^\  =  \MTt  i=l,  (a) 

or  log  i  —  ^Mm, 

•which  is  substantially  the  same  as  the  equation  given  on  page  286  ;  but  we 
could  not  assume  the  value  there  given  in  this  place,  for  it  was  found  by 
means  of  the  formulas  which  we  are  not  seeking.     Now  find  the  value  of 

20 

e  ^  given  on  page  278,  and  to  avoid  confusion  in  regard  to  letters  and  to 
simplify  the  expression,  we  will  use  the  equally  general  one 

it 
i  being  an  indeterminate  unit-vector,   V—  1. 

Take  log  i  =  log  i, 

passing  to  exponentials,  t  =  e°^  , 

raising  to  cp  power,  i*  =  e*  '^^  * 

and  by  Eq.  (a),  =  e*^*-*.  (&) 

2nB 


Making  g)  =  —-,  we  have 


•  See  advanced  works  on  the  Calctdus.     (Conrtenay's  Calculus,  p.  52.) 


328  APPENDIX. 

the  left  member  of  wMch  is  a  general  expression  for  n  rotations  througli  an 
angle  6  ;  hence,  (379,  (5)), 

%  "  =cos  «9  +  i  sin  nO  =  e^^iei^  ^^ 

also  cos  nB  —  i  sin  nii  —  e~^"'^^;  (e) 

adding,  cos  nQ  =  ^(e^^"^^  +  e-  ^^''^^),  (J) 

subtracting,  sin  nQ  =        (e^'»^*  -  g"  ^'^^«). 

In  the  last  two  equations,  let  ilf  =  1,  n=l,  i  =  V  —  1,  and  we  have 

^^  ®  =  i^l  (e'^^^  -  e-  ^v'^),  (A) 

which  are  known  as  Evler's  formulas. 

■  In  equation  (6)  let  cp  =  i,  and  observing  that  *^  =  —  1,  we  have 

which  in  the  Napierian  system  becomes  (since  M  will  be  unity) 


and  restoring  the  value  of   i  =  V  —  1,  we  have 
or  squaring,  .   (—  1)  =e    ^, 

—  y/—  1  n- 

taking  reciprocals,  (—1)  = «  »  (*») 

or  squaring  again,  1=6,  (n) 

which  equations  express  a  peculiar  relation  between  7t  and  e.  Equation  (n) 
at  first  sight  appears  to  be  incorrect  for  arithmetically  log  1  =  0;  but  upon 
examination  it  will  be  found  that  this  is  a  special  and  not  a  general  value, 
and  that  it  does  give  correctly  one  of  the  imaginary  values  of  log  1.  From 
it  we  find 

-  '^^=l  log  1  =  2tc; 

.•.logl  =  -;^^ 

=  2;r  V^^.  {o) 

From  equation  (d)  we  find 

log  (cos  nQ  +  V—1  sin  nQ)  =  MnQ  V^^. 


APPENDIX.  329 

Let  f)  ~Tt,  then  cos  nO  =  1,  and  sin  tiO  =  0,  for  all  integer  values  of  n,  and 
we  have 

log  1  =  Mrnt  V—  1* 

which  gives  the  real  as  well  as  inaaginaiy  values  of  log  1.     If  »  =  0,  then 

log  1  =  0, 

which  is  its  only  arithmetical  value,  and  which  also  is  independent  of  the 
system  in  which  it  is  taken. 

UM^laiidn  =  l  

log  1  =  ;r  ♦'-  1, 

which  is  one  of  the  imaginary  values  of  the  logarithm. 
If  Jf  =  1  and  »  =  2, 

logi  =  271  r^i, 

which  is  the  value  found  in  equation  (o). 

We  see  then,  that  in  Euler's  formulas,  De  Moivre's  formula,  and  in 
the  imaginary  exponentials  involving  tt,  the  imaginary  V—  1  of  algebra 
becomes  the  same  as  the  real  indetekminate  unit-vectok  of  quaternions. 

*  Professor  Graves  gave  a  more  general  equation,  thus, 
in  which  oa  and  co'  are  independent  integers.     (Phil.  Trans.,  1829.) 


APPENDIX  IL 


HYPEBr-SPACE. 

Prom  any  point  in  space  a  line  may  be  drawn  in  any  direction ;  hence  if 
every  such  line  were  considered  a  dimension  of  space,  the  latter  womd  have 
an  unlimited  number  of  dimensions,  and  the  expression  "space  of  n  dimen- 
sions "  would  be  perfectly  rational.  But  the  dimensions  of  space  are  not  so 
determined.  A  right  line  is  conceived  as  determining  one  dimension,  and 
space  confined  to  one  right  line  is  called  space  of  one  dimension.  If  from 
any  point  of  this  line  a  perpendicular  be  drawn,  then  will  the  space  limited 
to  the  plane  of  these  lines  be  called  »pace  of  two  dimensioiis.  Similarly,  if 
through  the  common  point  of  the  two  lines,  a  third  be  drawn  perpendicular 
to  the  former  ones,  any  point  will  be  determined  by  its  distance  from  the 
planes  of  the  lines,  taken  two  and  two,  and  such  space  is  called  space  of  three 
dim.ensions.  No  more  than  three  mutually  perpendicular  lines  can  pass 
through  a  point;  hence  higher  orders  of  space,  as  ihQ  fourth,  fifth,  nth,  etc., 
are  imaginary;  in  other  words,  have  no  real  existence.  Tri -dimensional 
space  is  the  highest  order  of  which  we  have  any  knowledge — it  is  natural 
space.  As  this  mode  of  determining  the  dimensions  of  space  dates  at  least 
from  Euclid,  it  is  sometimes  called  Euclidean  space  to  distinguish  it  from 
the  lower  orders  of  real  space,  and  the  higher  orders  of  imaginaiy  space,  and 
all  the  higher  orders  may  be  classed  as  hyper-space. 

The  solution  of  f  /oblems  involving  hyper-space  depends  upon  the  laws 
and  conditions  assigned.  It  generally,  however,  falls  under  the  following 
principle,  viz. :  The  laws  and  formulas  applicable  to  the  solution  of  problems 
having  real  conditions  may  be  extended  to  those  involving  unreal  conditions. 
This  principle  we  will  Ulustrate  by  a  few  examples  from  different  branches 
of  mathematics. 

Beginning  with  arithmetic,  we  take  the  simple  problem — If  one  and  one 
are  three,  what  wiU  two  and  two  be?  Now  one  and  one  are  two  and  nothing 
else,  and  in  the  nature  of  things  can  be  nothing  else  so  long  as  language 
retains  its  present  meaning.  We  are  aware  that  it  has  been  asserted  that  if 
we  were  in  some  part  of  the  universe  where  in  the  process  of  putting  one 
thing  by  (or  with)  another,  the  result  was  always  three,  we  would  admit  that 
one  and  one  make  three.  But  this  is  a  misstatement  of  the  case.  In  arith- 
metic, one  and  one  do  not  make  anything — they  are  two.    It  is  a  statement  of 


EYPEB-SPACE. 


331 


a  fact.  Everywhere  in  the  universe  one  and  one  are  two,  and  two  and  two  are 
four.  Hence  our  problem  may  read — if  two  be  three,  what  will  four  be? 
The  problem  involves  unreal  conditions;  but  by  assuming  that  the  same 
ratio  exists  between  four  and  the  required  result  as  between  two  and  three, 
the  problem  may  be  solved  by  proportion,  giving  six  as  the  result. 

By  introducing  different  units  into  this  problem,  the  unreal  conditions 
may  be  removed  ;  thus,  il  two  apples  cost  three  cents,  what  will  four  apples 
cost? 

Again,  if  a  melon  be  worth  20  cents,  and  it  be  divided  into  two  equal  parts 
so  that  one  part  shall  be  twice  as  large  as  the  other,  what  will  each  part  be 
worth?  Here  the  conditions  are  seK-contradictory,  and  there  is  no  rational 
solution ;  still,  by  assuming  that  ^  is  ^,  then  by  proportion  10  cents  will  be  65 
cents;  then  the  other  half  will  be  two-thirds;  and  the  cost  183  cents.  But 
by  a  slight  change  in  the  wording  the  contradictory  character  is  removed ; 
thus,  if  one  half  is  worth  twice  as  much  as  the  other,  what  will  each  half  be 
worth? 

In  trigonometry  the  cosine  of  a  circular  function  cannot  exceed  radius  or 
unity  ;  but  if  it  be  assumed  that  cos  cc  =  3,  a  value,  or  rather  an  expression, 
may  be  found  for  each  of  the  other  trigonometrical  functions  by  assuming 
that  the  same  law  holds  for  the  unreal  as  for  the  real  conditions.  Thus 
we  would  have  for  the  sine,  sin  x=  4  (1  —  cos-  x)  =  |/  (1  —  4)  =  ^  —  3  = 
y  3  4/  —  1 ;  a  result  which  shows  that  sin  a;  is  impossible  when  cosx  =  2. 

In  geometry  the  square  of  the  hypothenuse  of  a  right-angled  triangle 
equals  the  sum  of  the  squares  of  the  two  sides.  In  a  rectangular  parallelo- 
piped  the  square  of  the  diagonal  joining  the  opposite  angles  equals  the  sum 
of  the  squares  of  the  three  adjacent  edges.  If,  therefore,  a  solid  be  constructed 
upon  four  (imaginary)  mutually  perpendicular  lines  as  edges,  the  square  of 
the  diagonal  through  the  extreme  angles  will  equal  the  sum  of  the  squares 
of  the  four  adjacent  edges,  and  hence,  if  the  length  of  each  edge  be  unity, 
the  length  of  the  diagonal  would  be  2. 

Questions  pertaining  to  space  of  n-dimensions  is,  as  has  been  intimated, 
purely  an  extension  of  analysis  from  certain  real  to  certain  imaginary  cases ; 
and  to  illustrate  this  stUl  further,  we  make  use  of  the  analysis  of  coordinate 
geometry. 

Tabulating  the  well-known  results  for  space  of  one,  of  two,  and  of  three 
dimensions  in  terms  of  rectangular  coordinates,  we  have 


No .  of  Dimensions 
of  Space. 

Equations  to  a 
Point. 

Eqnatlons  to  tlie 
Right  Line. 

Equations  to  the 
Plane. 

1 
2 
3 

X—  a=0 

x  —  a  =  0 
y-b  =  0 

X  —  a  =  0 
y-b  =0 
z  —  e  =  0 

X  indeterminate 

X  +my  —  a  =  0 

X  +  mz  —  a  =  0 
y  +  nz  —  b  =  0 

N(me 

X  and  y  indetermi- 
nate 

X  +  Ay  +  Bz  + 
2)  =  0 

332 


APPENDIX  n. 


In  this  table,  we  observe  that  the  equations  to  a  point  are  of  the  first  degree 
of  two  terms  each,  and  the  same  in  number  as  the  corresponding  coordinates 
in  space.  The  equations  to  a  right  line  are  also  of  the  first  degree  between 
two  variables,  and  one  less  in  number  than  the  corresponding  ones  for  a  point, 
and  the  equation  for  a  plane  is  of  the  first  degree  between  three  variables. 
We  also  observe  that  the  form  of  the  equations  to  the  right  line  may  be  found 
by  adding  the  last  of  the  equations  to  a  point  to  each  of  the  preceding  ones 
separately ;  thus  z  —  c  added  iox  —  a  gives  x  +  z  —  a  —  c  —  0,  which  is  gen- 
eralized in  the  form  x  +  mz  —  a  =  Q;  and  similarly  for  the  others.  Pro- 
ijeeding  in  this  way  the  following  table  is  formed  : 


Number  of 
rectangu-  ' 
lar  axes  or  Equations 

so-called    to  a  Point, 
dimensions 

of  space. 


X  -a  =  0 

X  -  a  =  0 
-6=0 

-  a  =  0 
y  -b  =  0 
z  -  c  =  0 

w-  a  =  0 
-b=0 
y  -  c  =  0 
z  -rf=0 

V  -  a  =  (i 

w-  6  =  0 

X  -  c  =  0 

y-  d  =  0 

z  -  e=Q 


X  indeterminate 

x  +  mz  —  a=0 


Equations 

to  tbe 
Bight  Line. 


Equations  to  the 
Plane. 


None 

None 

~_i  •-,-     n  —  n'  x,y,&n(lz  in- 

yf^l'b  =  0  f:+^y+^''+G=0  determinate 


None 

X  and  y  indetermi- 
nate 


Te^Hyper^  Equations  to  the 
nlanpofthP     Hyper-plane  of 
do?dlr     Ithe  second  order. 


None 
None 

None 


w+  Iz  -a-0  '^_^^„  1  53  1    c=(^  w-\-Ax-^By\w,x,y,&nAz  w 
x  +  wz-0-0  a,^2^^jg-2J).j?=o|+  Cz-\-D  =  Q^determinate 


v-\-kz  -  a  =  0 
w-\-  Iz  -  6=0 
x-\-mz  -  c=0 
y  +  m  —  d  =0 


.\v  +  Ax  +  By[ 


v4-Ay  +  Bz+C=0\\'y<,^f.t  =  lh  +  Aw  +  Bx 


\+  Gz  +  JI=0 


—  n— U 


In  a  similar  manner,  the  equations  to  the  hyper-solids  may  be  written. 
Thus,  since  aU  spheres  wUl  have  a  constant  radius,  we  have  for  the  hyper- 
sphere  of  the  first  order,  or  the  sphere  of  four-dimension  space,  the  equation 

«^  +  «»  +  2/»  -+-  3"  =  r". 

Similarly,  the  equation  of  the  hyper-eUipse  of  the  first  order,  the  origin 
being  at  the  centre,  wiU  be 

M»«      a~»      «»       2»       .      . 


a",  J',  c",  <?"  being  positive.  For,  the  section  of  each  of  the  coordinate 
planes  with  this  surface  wiU  be  an  ellipse,  to  determine  which  eliminate  two 
of  the  variables,  as  w  and  x  for  instance,  by  means  of  the  two  equations  of 
the  plane  of  four-dimension  space.  If  the  equation  of  the  hyper-plane, 
w  +  Ax  +  By  -f  Cz  -f  2>  =  0,  be  combined  with  the  preceding  equation,  only 
one  of  the  variables  can  be  eliminated,  and  the  intersection  vriU  be  given  in 


HYPER-SPACE.  333 

terms  of  three  variables,  and  in  order  to  discuss  the  properties  of  the  curve 
in  this  case,  it  will  be  necessary  to  determine  the  general  properties  of  hyper- 
curves  referred  to  hyper-coordinates.  It  is  not  our  purpose  to  develop  the 
subject  further. 

Another  view — a  moving  point  will  generate  a  line;  a  moving  line  may 
generate  a  surface ;  a  moving  surface,  a  solid ;  hence,  a  moving  solid  might 
generate  a  hyper-solid,  and  so  on. 

Some  amusing  problems  have  already  been  stated  involving  four  rectangu- 
lar coordinates.  In  the  American  Journal  of  Mathematics  vol.  i.,  p.  1 
(1878),  it  is  shown  that  a  spherical  shell  in  four-dimension  space  may  be 
turned  inside  out  without  tearing  or  stretching;  and  in  the  .same  Journal, 
voL  iii.,  p.  1,  several  regular  solids  in  four-dimension  space  are  determined 
and  named — thus  affording  another  instance  that  the  mathematician  may 
create  (or  imagine)  that  which  is  impossible! 


APPENDIX  III. 


To  find  the  distance  between  two  lines  in  space.    Let  the  lines  be 
X  —  mz  +  a  )  x  =  m'z  +  a'  ) 

^    (1)  h  (3) 

y  =  m  +  b    )  y  —  n'z  +  b'    ) 

and  pass  a  plane  through  (1)  parallel  to  (2)  ;  then  will  the  required  distance 
be  the  length  of  the  perpendicular  between  (2)  and  the  plane  thus  passed.  To 
determine  the  plane,  draw  a  line  through  any  point  of  (1)  parallel  to  (2)  ;  and 
for  convenience  take  the  point  where  (1)  pierces  the  plane  xy,  which  point 
will  be 

X  =  fi,  y  =  b,  s  =  0  ;  (3) 

and  the  line  through  this  point  parallel  to  (2)  will  be  (Art.  197,  Eq.  (2) ), 

X  —  a  =  m'z  ) 

[.  (4) 

y  —  b  —  n'z   ) 

This  line  pierces  the  plane  xz  in  the  point 

y^O,        z=-^,       ^  =  «--^:  (5) 

and  (1)  pierces  the  same  plane  in  the  point 

y  =  0,     ■  z  = ,        X  =  a .  (6) 

"        '  n  n 

The  three  points  (3),  (5),  (G),  will  determine  the  plane.     The  equation  of 
the  plane  being  of  the  form  (Art.  202), 

Ax  +  By  +  Cz  +  D  =  0,  (7) 

gives,  with  (3),  (5),  (6),  the  equations  of  condition, 
334 


APPEXDIX. 
a4  +  65  +  0  +  Z)  =  0 


G-^0 


a -)  A  --^C  +  Z)  =  0 


s^         n  /  n 


335 


(8) 


from  which  we  find 


A  = 
B  = 
C= 


n 

—  n 

a  {n' 

-n) 

—  h{m' 

—  m) 

m' 

—  m 

a\n' 

-n) 

—  b{m' 

—  m) 

inn' 

—  m'n 

D  ^ 


(9) 


a  (ti'  —  n)  —  b  (rn'  —  m) 
which  in  (7)  gives  for  the  equation  of  the  plane 

{n  —  n')x  +  {rn'  —  m)y  +  {mn'  —  m'n)z  +  a  {n'  —  n)  —  b'  {m!  —  m)  =  0.   (10) 
The  equation  to  the  line  perpendicular  to  this  plane  from  the  point 

x  =  a',  y  -b',  zz=  0,  (11) 

where  (2)  pierces  the  plane  xy  will  be  (Art.  215), 


X—  a  =  -^  2, 


(13) 


and  the  point  where  this  line  pierces  the  plane  (7)  will  be  (Art.  212), 


.  a' A  +b'B  +  D 
Xi  =  —  A   ,..       „.,       ^.  +  a' 


„  a' A  +b'B+  D      . 
''=-^A^  +  B^  +  C^^^'      ■' 


(13) 


Zx  =  -C 


a' A  +b'B  +  D 


A^  +  B^  +  C* 
Hence,  the  required  distance  will  be,  Equations  (11)  and  (13), 


I-  V'(^'  -  «')*  +  (yi  -*')*  +  8i*, 


336  APPENDIX. 

which  reduced  by  the  aid  of  (13)  gives 


^^^TA^VB^D^  (14) 


(15) 


and  this  reduced  by  the  aid  of  (9)  gives 

7—       (a  +  a')  (n  —  n')  +  (b  —  b')  {m  —  m') 
^{m'n  —  mn')''  +  (wl  —  ntif  +  («-'  —  n)s 

If  0  be  the  angle  between  (1)  and  (2),  we  have,  Art.  199,  Eq.  (6), 

cos  0  = — ;  (16) 

y'w^  +  n''  +  1  .  /y/m'*  +  ft'*  +  1 


.•.  sin  S  =  V^wi'w  —  «i»')*  +  {rn'—mY  +  (»'  —  %)* 
Y^m*  +  «,='  +  1  .  '^m'^  +  »'^  +  1 

and  hence  (15)  may  be  written 

,_      {a  4-  a')  (n  —  n')  +  (b  —  b')  (m  —  m') 
sin  (3  /y/m^  +  re*  +  1  .    /y/w'^  +  w"*  +  1 


(17) 


(18) 


If  one  of  the  lines,  as  (2)  for  instance,  passes  through  the  origin  of  co-or- 
dinates, we  have  a'  =  0,  and  6'  =  0,  and 


, a{n  —  n')  +  b{m  —  m') 


sin  0  ^/m^  +n^  +  \  .   /y/m'^  +  n'^  +  1  ' 
and  further,  if  (2)  is  in  the  plane  xy,  we  have  m'  =  0,  and 


(19) 


1  = a{n-n')-^bm_  .  ^^^ 

sin  e  -y/ot*  +  to«  +  1  .  -v/»'*  +  1 

and  still  further,  if  the  line  (2)  coincides  with  the  axis  of  (g),  we  also  have 
n'  =  0,  and  (15)  becomes 

^^jm+Jm^  (21) 

If  lines  (1)  and  (2)  are  parallel,  eqtiation  (15)  reduces  to  ^r^ ,  as  indeedwe 


APPENDIX. 


337 


might  have  anticipated,  since  by  our  mode  of  constructing  a  line  through  any 
point  of  (1)  parallel  to  (2)  would,  in  this  case,  coincide  with  (1),  an  unlimited 
number  of  planes  could  be  passed  parallel  to  (2).  It  will,  therefore,  be  neces- 
sary to  make  an  independent  solution  of  this  case. 


DISTANCE  BETWF,EN  TWO  PARALLEL   LINES. 
Let  the  lines  be 

x  —  mz-\-a)  x  =  rm  +  a' 

\  ;      (33) 

y  =  m  +  b    )  y  =  m  -\-  h' 


(23) 


Line  (23)  pierces  plane  s-y  in  the  point  {a',  b',0),  from  which  pass  a  perpen- 
dicular to  (22)  ;  its  equation  will  be  of  the  form,  Art.  197, 


X  —  a'  =  m"z 
y  —  b'  =  n"z 

and  the  equation  of  condition  of  perpendicularity  will  be,  Art,  199, 

mm"  +  nn"  +  1=0, 

and  the  condition  that  (24)  and  (22)  intersect  will  be,  Art.  196, 

a'  —  a        V  —  h 


m  —  m"      n  —  n' 


(24) 


(25) 


(26) 


Prom  (25)  and  (26)  we  find 


„  _  (&'  —b)mn  —  (a'  -  a)  (n^  +  1) 
~"         {b'  —  b)n+  {a'  —  i)m 

„__  —  (b'  —  h)(m^  +  1)  +  {a'  —  n)mn 
~  (6'  —  6) «.  +  (a'  —  a)m 


r ; 


(27) 


which  in  (24)  will  give  the  equations  of  the  perpendicular,  and  eliminating 
between  that  and  (22),  the  point  of  intersection  of  (22)  and  (24)  will  be  found 
10  be 


(6  —b)n  +  (a  —  a)m 

Xi  =  ~ — -\, r^ —  m  +  a 

m^  +  n^  +  1 


y> 


_  (p'  —  l)n  +  (a'—a)m 
~         m'^  +  n"  +  1 


n  +  b      - 


—  (^'  —  ^)  ^  +  (a'  —  a)m 


(28) 


338  APPENDIX. 

The  required  distance  will  be 


d  =  ^{Xi  -  a')-  -h  {y,  —b')'+  s" 


7(6'  -  by  (m"  +  1)  -  2(5'  -  &)(«'  -  a)  /ran  +  (g'  -  ay  (»«  +  1) 


If  one  of  the  h'nes,  as  (23),   passes  through  the  origin,  a'  and  b'  will  be 
zero,  in  which  case  we  have 


a  =  i/^^ ^"^'  -e  1)  -  2a6 .  mra  +  a-  (n^  +  1)  .^q. 


and  also  if  the  lines  be  parallel  to  the  plane  xz,  we  have  «  —  0,  and  the 
equation  becomes 


^-v'^'VlV'  ™ 


still  lurther,  if  the  lines  be  in  the  plane  xz,  b  will  also  be  zero,  and 


d  = .  (32) 

^/m'  + 1 

If  the  lines  be  in  a  plane  parallel  to  xz,  and  neither  passes  through  the 
origin,  we  have  n  =  0,  and  6  =  b',  and  (29)  reduces  to 


d  =     ^  "^    .  (33) 


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